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Autonomous Systems Lab Prof. Roland Siegwart Studies on Mechatronics Spherical Rolling Robots Autumn Term 2013 Supervised by: Author: Péter Fankhauser Jan Carius Marco Hutter Declaration of Originality I hereby declare that the written work I have submitted entitled Spherical Rolling Robots is original work which I alone have authored and which is written in my own words. 1 Author(s) Jan Carius Student supervisor(s) Péter Fankhauser Marco Hutter Supervising lecturer Roland Siegwart With the signature I declare that I have been informed regarding normal academic citation rules and that I have read and understood the information on 'Citation eti- quette' (https://www.ethz.ch/content/dam/ethz/associates/students/studium/ exams/files-en/plagiarism-citationetiquette.pdf). The citation conventions usual to the discipline in question here have been respected. The above written work may be tested electronically for plagiarism. Place and date Signature 1 Co-authored work: The signatures of all authors are required. Each signature attests to the originality of the entire piece of written work in its nal form. Contents Abstract v Symbols vii 1 Introduction 1 2 Mechanical Principles 3 2.1 Mechanics of the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spinning Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3.2 Advantages and Disadvantages 2.4 2.5 Reaction Wheels . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Advantages and Disadvantages . . . . . . . . . . . . . . . . . 3.2 Centre of Mass Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5.2 Advantages and Disadvantages 17 . . . . . . . . . . . . . . . . . Overview 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 Robot by University of Science and Technology Kraków . . . 19 3.1.2 Sphero 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1.3 RoBall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.4 Spherical Rolling Robot by University of Delaware . . . . . . 22 3.1.5 Spherical Mobile Robot by Indian Institute of Technolgy . . . 22 3.1.6 Robot by Massachusetts Institute of Technology . . . . . . . 23 3.1.7 Gyrover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.8 Gyrobot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.9 Reactobot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.10 Robot by Kobe University . . . . . . . . . . . . . . . . . . . . 28 Conclusions from Existing Designs 28 . . . . . . . . . . . . . . . . . . . 4 New Design Ideas 4.1 8 11 2.5.1 3 Existing Designs 3.1 3 31 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.1.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.1.2 Fullling the Desired Requirements . . . . . . . . . . . . . . . 32 4.2 Omnidirectional Car-based . . . . . . . . . . . . . . . . . . . . . . . 4.3 Complicated Pendulum with Flywheel . . . . . . . . . . . . . . . . . 34 4.4 Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5 Conclusion 5.1 Outlook 33 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii 39 A Gyroscope Dynamics 43 B SimMechanics Model 45 Bibliography 48 Abstract This report summarizes our work on the analysis and design of round rolling robots. This includes spherical as well as single wheel vehicles which are fully contained in their round shell. A mechanical analysis and modelling of dierent locomotion techniques show that there are three main principles to achieve motion with these type of robots: 1. Using the large angular momentum of gyroscopes for stabilization and torque amplication, 2. manipulating the internal angular momentum by means of reaction wheels and 3. taking advantage of a gravitational induced torque when the centre of gravity lies outside the contact point. We show that a gyroscope is capable of producing the largest torques, however a centre of mass shift is superior when it comes to supplying a constant torque over a longer period of time. Reaction wheels are suitable for storing and releasing momentum. Since all of these abilities are usually desirable, a combination of two or more of the above mechanisms seems promising for a highly agile vehicle. We summarize and compare ten robots that have already been developed. Most of them use more than one propulsion principle, the most popular way to generate torque is by shifting of an internal mass and this is almost always realized by a hanging pendulum mass being pitched in the desired direction. We develop a new idea where a pendulum mass is used dynamically as an excentric reaction wheel. Such a design is proposed in the last section and could be the basis for further research on this topic. v Symbols Symbols Ω Absolute angular velocity of a rigid body L Absolute angular momentum of a rigid body T Torque I, K eIx Ca Inertial- resp. body-xed coordinate system Unit base vector of the I system in the Coordinate representation of vector a in ABI Coordinate transformation matrix from ωIC Angular velocity of system g C x-direction C I coordinates to B system relative to system I Acceleration due to gravity Vectors and Matrices are denoted in bold (x) and scalars in light (h). Acronyms and Abbreviations ETH Eidgenössische Technische Hochschule RW Reaction Wheel CG Centre of Gravity DoF Degree of Freedom vii Chapter 1 Introduction Land robots may have dierent concepts of locomotion, for example walking on two or more legs, driving by means of wheels or as a track vehicle, hopping or rolling on parts of the robot or its entire surface. Although nature often resorts to legged systems because of the diculty of rotating joints, most technical applications use rolling as the primary way of locomotion. Rolling has the advantage of being very ecient on a variety of dierent terrains. Furthermore dicult step length calculations and stability issues [1] can be avoided. Of special interest in this report is the branch of rotund robots. In this context, we dene a spherical robot to be a machine, whose outer surface has the shape of a sphere, i.e., there are no extremities and the whole interior is contained in a spherical hull. Such systems are called encapsulated. This design appears inter- esting, since there is a priori no preferred orientation which means the robot can move omni-directionally and the outer hull can be entirely sealed to provide protection to the interior parts. It is self-evident, that rolling will be the main mode of locomotion, however this does not exclude any hopping manoeuvres or the like. Another advantage of a spherical shape is that all positions can be statically stable, allowing the robot to rest on any point of its surface. Further investigation and design considerations will show if this feature can be maintained when the internal mechanisms come into play. In the past, propulsion by means of a controllable pendulum or hamster-wheel designs have enjoyed great popularity [2]. For exploration purposes in large and at environments, wind-driven concepts have also been mentioned, however in this work we require much more active steering and acceleration capabilities. The motivation of this study is to nd out what manoeuvres round robots are potentially capable of. The goal of this report is to present a number propulsion mechanisms available to round robots and to identify concepts for high performance manoeuvres. We want to explore what devices and mechanism can be employed to provide extra capabilities such as fast acceleration and jumping over steep obstacles. In chapter 2 the underlying mechanical properties will be analysed and modelled and advantages and drawbacks shall be considered. Subsequently chapter 3 will give an overview over several previously existing round robots that are documented in literature. They are ordered according to their locomotion method. In chapter 4, requirements for a new design will be presented and conclusions from the previous chapters are drawn to propose new ideas and designs for a round robot. ideas will be analysed and modelled. Several Chapter 5 concludes with an outlook for further work on this topic and how development may be continued. 1 Chapter 1. Introduction 2 Chapter 2 Mechanical Principles This chapter provides the basic mechanical analysis for dierent propulsion strategies. This is to explore a variety of options how locomotion of a spherical robot can be achieved and what advantages and disadvantages the concepts have to each other. As for the modelling, all linear and angular velocities as well as linear and angular momenta are given with respect to an inertial (world) reference frame. Unless otherwise stated, all bodies in this text shall be considered rigid bodies and friction is neglected. The latter especially applies to all joints and couplings which are considered to be ideal. Before the specic systems are considered, we will at rst take a look at spherical and spinning bodies in general. 2.1 Mechanics of the Sphere A sphere in space (as any unconstrained body) has six degrees of freedom (DoF); three translational and three rotational. When there is ground contact extra couplings need to be considered. Assuming no slipping and no taking o, the DoFs are reduced to three: Two for motion (rolling) along the plane and one for the yaw axis (turning in place). If the contact point is perfect, yaw rotation is only needed for reorientation in case the device is not omnidirectional. Usually, we are only interested in planar motion and hence in theory two independent actuators should be sucient to control both relevant DoFs independently. Das and Mukherjee showed that by using a discontinuous control strategy one can achieve global stability and exponential convergence of an arbitrary conguration for a sphere [3]. The control inputs must be chosen in a rotating frame of refer- ence which result in independent sweep and tuck manoeuvres. Sweep manoeuvres make the sphere go along a circular arc (xed distance to a point on the plane) whereas tuck manoeuvres make the sphere move in a radial direction towards or away from the same point. Therefore two actuators, if positioned correctly, are in theory enough to make the rolling sphere fully controllable. In fact, there are certain designs that only use two motors. 2.2 Spinning Devices Spinning devices can be grouped into dierent categories. For the remainder of this text, the following denitions are used: 3 Chapter 2. Mechanical Principles 4 Figure 2.1: Simple Disc Flywheel A ywheel is a rotating device used for energy storage. It is often used to ensure a continuous energy supply for discontinuous motors by providing a uniform rotational speed through its high inertia. In this text, a ywheel is simply a spinning disc. Gyroscope A gyroscope is an attitude control device. The spinning disc is mounted in gimbals, which allow rotational motion. The number of DoFs available depends on the specic setup and usually at least one DoF is actuated and can be tilted. Through a large angular momentum, small input torques get amplied and lead to large output torques in a perpendicular direction. Usually the rotor speed is held constant and large torques can be produced without excessive power consumption. Reaction Wheel A reaction wheel (RW) is a device that is spun around a xed axis. By accelerating it, a motor reacts against it thereby producing a torque on its stator that is mounted to an external structure. By changing the rotational speed it can therefore change the orientation of the robot and may release a lot of energy very quickly that has been accumulated over a longer period of time. D, R, thickness b and constant density ρ. Its centre of introduce a coordinate system B as drawn in gure In order to simplify further modelling, we will limit ourselves to a spinning disc i.e., a short cylinder with radius gravity is denoted with S. We 2.1. The mass of this disk is m = ρV = ρbπR2 . (2.1) In the following, we will often use the inertia tensor, which will be derived below. All integrations are done with respect to the B -system. By denition the tensor of inertias is R B ΘS = (y 2 + z 2 )dm RD −(xy)dm −(xz)dm D D R R −(xy)dm 2 (x + z 2 )dm D R −(yz)dm D R D R RD −(xz)dm R D −(yz)dm 2 2 (x + y )dm D (2.2) 5 with 2.3. dm = ρdV . Gyroscopes By using polar coordinates to compute these volumetric integrals, one nally arrives at B ΘS which shows that our br R2 4 = m 0 0 B -system 0 + 0 b2 12 2 R 4 0 0 + 2 b 12 (2.3) consists of three principal axes. For very thin disks this can be further simplied to B ΘS 2.3 R2 2 =m R2 · diag(2, 1, 1) . 4 (2.4) Gyroscopes 2.3.1 Modelling Gyroscopes are interesting to study in the aspect of dynamics and control because they inherently exhibit some unintuitive behaviour. They are highly agile and powerful, however also dicult to control and handle. In order to understand their behaviour, we take a general approach: All bodies are reluctant to changes in their rotational momentum. If a body is at rest and a torque acts on it, the body will begin to rotate in the direction of the torque. In this simple case, the relation between torque T and rotational acceleration ϕ̈ is given by ϕ̈ · Θ = T where the constant of proportionality Θ (2.5) is called moment of inertia (along the axis of the torque or rotation). The moment of inertia plays a similar role as mass plays in translational motion. The peculiar properties of a gyroscope arise when the object subjected to a torque is already spinning. In this case, the principle of angular momentum for rigid bodies has to be considered to capture all eects. It links the time derivative of the angular momentum MO L̇O of a body with respect to a stationary point O to the external torque O: acting on the same body also expressed with respect to the same point L̇O = MO . (2.6) In basic terms, this equation states that the angular momentum vector will gradually move (or increase) in the direction of the applied external torque. This results in some very unintuitive behaviours if the angular momentum is not parallel to the applied torque. The full derivation of the equations of motion of a gyroscope can be found in the appendix under A (page 43). The following subsections only present the most important ndings. Precession Precession is an inherent property of gyroscopes when subjected to external torques. Mathematically it is described by T =Θ·ω×Ψ. T is ω the where the torque on the gyroscope, axis, spin angular speed and Ψ Θ (2.7) the moment of inertia about the spin the wheel precession rate. Chapter 2. Mechanical Principles 6 Figure 2.2: Single gimbaled gyroscope Using a simulation of a spinning disc as developed in Appendix A (see (A.9)) this behaviour can be clearly seen. For a ywheel spinning at a high angular velocity around its axis of revolution (x-axis in this case), an input torque in the rection makes the wheel turn around the z -axis. y di- The precession rate is inversely proportional to the spin speed. This eectively means that the spinning gyroscope does not `retreat' in the direction one would expect a stationary body to do, but it actually turns around an axis perpendicular to both the torque and the spin axis. This could be exploited for example by using an input torque engendered by gravitational forces to produce an output torque without reaction on the sphere. An alternative approach to produce torques without reaction on the hull is to use two counter-rotating ywheels that can react against each other for tilting. If the gyroscope encounters resistance (for instance damping forces or rigid joints) in the direction of precession, these resistive torques make it eventually turn in the direction that one would intuitively think of. In this case, large torques will be experienced by the suspension. An analysis of this phenomena is provided in the following section. Torque Amplication One reason for using gyroscopes in the propulsion of robots (and also aerial or space vehicles) is their torque amplication ability. With a relatively small motor input torque, large torques can be applied to the system. This property will be derived in the following. Consider gure 2.2 where ϕ describes the spinning motion and α accounts for the turning motion of the gimbal. There are three coordinate systems to realize this motion. The system G has its origin at around the common S I system is inertially xed in space; the intermediate (the centre of gravity (CG) of the disk) and is rotated x-Axis of I and axis of symmetry of the disk; the the common z -Axis of G and B. G Evaluating all equations in the B G by the angle α such that eG z is parallel to the system is xed to the disk and rotated around system is easiest in this case. The total angular velocity can easily be deducted from the schematic and is given by GΩ The G = G ωIG + G ωGB α̇ = 0 . ϕ̇ (2.8) system is a principle coordinate system and the moment of inertial tensor with respect to S is constant in time when expressed in G. Due to symmetry the 7 2.3. Gyroscopes rst two entries are the same and hence ΘA 0 G ΘS = 0 0 ΘA 0 0 0 . ΘB (2.9) It is straightforward to calculate the angular momentum of the disk as a function of α and ϕ and their derivatives. Using Euler-Dierentiation one can nd the derivative of the angular momentum to be ΘA α̇ 0 , G LS = G Θ̄S · G Ω = ΘB ϕ̇ ΘA α̈ −ΘB α̇ϕ̇ . G (L̇S ) = ΘB ϕ̈ Now the principle of angular momentum can be applied in the G (2.10) system with M being the external torque on the disk which reads MxG ΘA α̈ −ΘB α̇ϕ̇ = MyG . ΘB ϕ̈ MzG G (L̇S ) = GM ⇔ It is apparent that a torque in the speed of the disk. const , ϕ̈ = 0. (2.11) z -direction is only required to change the spinning ϕ̇ = In a pure gyroscope this us usually held constant, hence This simplication leads to MxG ΘA α̈ −ΘB α̇ϕ̇ = MyG . 0 MzG (2.12) z -component shows that no torque is We can conclude several things: First, the required to keep the disk spinning (ideal frictionless case) because the other two ϕ̈. Second, we look at how the input torque τin = MxG = I Mx relates to the output torque τout = MyG (which will be the torque on the whole structure). Using the initial conditions that the gimbal is at rest and the gyroscope torques cannot inuence spins at a constant speed ϕ̇ 6= 0 α = α̇ = 0 (2.13) it follows from (2.12) that τout (t) = MyG (t) −ΘB ϕ̇ = −ΘB ϕ̇α̇ = ΘA Zt τin (t̂) dt̂ . (2.14) 0 (2.14) matches the result by Brown and Peck [4] calculated for control moment gyroscopes: Tcmg = −φ̇ · g × hr where Tcmg is the output torque, gimbal frame), g φ̇ (2.15) the gimbal rate (i.e., the rate of rotation of the the direction of the gimbal axis and hr the rotor momentum. Eectively a gyroscope `accumulates' and amplies the input torque and changes its direction. This means an input torque lets the gimbal rotate even if it was only applied for a limited amount of time. As long as the gimbal rotates (i.e., the integral 6= 0) a torque acts on the structure. This torque rotates with respect to an inertial frame of reference because it turns with the gimbal. Therefore the gimbal assembly does not provide a torque in a constant direction which may potentially make it dicult to control. The amplication is proportional to the spinning speed of the ywheel. The ratio of the moment of inertias is constant and for a thin disc is ΘB /ΘA ≈ 2 (from (2.4)). Chapter 2. Mechanical Principles 8 The kinetic energy only changes insignicantly during torque amplication and the power required for a manoeuvre is ideally only the useful output power [5]. In these terms a gyroscope is more ecient than a reaction wheel but it largely depends on how much power is lost just by maintaining a constant spinning speed of the gyroscope disc when friction comes into play. 2.3.2 Advantages and Disadvantages Below, reasons for and against using a gyroscope are listed: Advantages Torques in all directions are possible when all three DoF of the gyroscope are actuated. It can therefore turn in a spot. Can produce very high torques due to torque amplication properties. Naturally resistant against changes in attitude, hence the robot gets stabilized and smooth motion can be expected. If mounted in a wheel instead of a sphere, gravity induced torques can be used to generate precession. Disadvantages Energy losses due to fast rotating disc even when not providing any torque. Dicult to control and to nd path planning algorithms; all DoFs become coupled. Cannot stand on a slope for innite time. Lots of energy is stored in the ywheel which might be dangerous if control fails. Torques are available only for a limited amount of time (until gyroscope becomes misaligned) and even during this time the torque will most likely not be constant. Gyroscope is not eective at all times. Bringing it back into desired position may induce undesired torques and takes time. 2.4 Reaction Wheels 2.4.1 Modelling A reaction wheel is a relatively simple device and may be used in conjunction with a motor to create a torque in one specic direction. It has the potential to amass energy over a longer period of time and release it very quickly as well as providing a constant torque for some time. In order to demonstrate the most important properties of a reaction wheel, a simple R, mass M , moment of inertia IS ) with a central axis onto which a reaction wheel (radius r, mass m, moment of inertia ID ) is mounted. A motor exerts a torque T on the 2D model (gure 2.3) shall be considered. It consists of a sphere (radius reaction wheel and the equal but opposite torque is felt by the sphere hull. A no-slip constraint with the ground is enforced. Both bodies have their centre of gravity at their common geometric centre S whose horizontal position is captured by angular orientation of the sphere is denoted with ψ, of the disk with ϕ. zS . The 9 2.4. Reaction Wheels Figure 2.3: Sphere with central reaction wheel For both rigid bodies, the principle of angular momentum with respect to S and the principle of linear momentum are set up: ID ϕ̈ = T , (2.16) IS ψ̈ = −T − Fground · R , (2.17) (m + M )z̈S = Fground . (2.18) Enforcing the no-slip condition ⇒ ϕ̇R = żS ϕ̈R = z̈S (2.19) gives us uncoupled linear second order dierential equations as a function of the driving torque T: ϕ̈ = z̈ = T , ID IS R (2.20) T . + (m + M )R (2.21) There are several things to note from these equations: First, it is clear (and also very intuitive) that the reaction wheel and the sphere accelerate in opposite angular directions. If both are initially at rest, they begin to spin in opposite directions which may create unfortunate gyroscopic eects as seen in section 2.3.1 on precession. It is desirable to keep ϕ as small as possible for reasons mentioned above. From this viewpoint we prefer to have a high moment of inertia ID of the disk. Another important thing to notice is that all second derivatives of the variables are proportional to the applied torque. This torque may be achieved by a motor or alternatively by braking, i.e., trying to synchronize the ywheel rotational speed with the one of the sphere. What we are really interested in is the horizontal dynamics (2.21), since this describes how the sphere can move along the plane. For fast acceleration capabilities, we would therefore like to have a small total mass inertia of the sphere IS . Since for a spherical hull (m + M ) and a small moment of IS ∝ R2 , a small R is benecial for large accelerations. Overall, it would therefore be favourable to have a large but thin inner disk with most of its mass at the outer edge and a very light spherical hull with small radius. Chapter 2. Mechanical Principles 10 Figure 2.4: Sphere on a slope The components that are xed to the sphere should be placed as close to the centre as possible. Finally, let us consider the case where there acts a counter torque on the sphere, for example through friction. In this case one has to replace the torque an eective torque Teff = Tmotor − Tlosses , T in (2.21) by i.e., the motor torque is simply reduced by the counter torque. In the specic example of climbing a slope (angle of incline γ, see gure 2.4), there is a gravity induced torque: Tgravity = (m + M )gR sin γ . If we let ξ (2.22) denote the position along the slope, we quickly nd the dynamics using (2.21) to be Teff − (m + M )gR sin γ ξ¨ = . IS R + (m + M )R (2.23) Instant Braking Mechanism Inspired by Cubli [6], a cube which can jump on its edge and autonomously balance on one edge or corner, ywheels have the potential to momentarily create very large torques. The idea employed in Cubli is that by using brakes to quickly decelerate fast moving ywheels, one can generate much higher torques than any motor of acceptable size and mass would be capable of: A motor slowly accelerates a ywheel up to given speed and keeps it at this angular velocity. When needed, brakes are applied and almost instantly release the angular momentum stored in the ywheel, i.e., transfer it to the whole system. This will not be very energy ecient because of the energy lost in the brakes but may enable our robot to escape from holes or similar places where large torques are needed. Such a braking mechanism is investigated in the following. The sphere is modelled similarly to the section above and is initially resting on at ground. The ywheel inside is spinning (ϕ̇init > 0) and P of has an angular momentum with respect to the xed ground contact point Linit = ID · ϕ̇init . p (2.24) If now brakes are applied such that the relative angular speed between disk and sphere is brought to zero (ϕ̇ = ψ̇ ) very quickly, a lot of angular momentum is trans- ferred from the disk to the hull. By using the conservation of angular momentum around P and the no-slip condition ψ̇R = v where v is the horizontal speed, we 11 2.4. Reaction Wheels nd: no − slip −−−−−−→ ! Linit p = ID · ϕ̇init = ⇔ = v Lfinal = ψ̇(ID + IS ) + (m + M )Rv , p v (ID + IS ) + (m + M )Rv , R ID ϕ̇init . ID +IS + (m + M )R R (2.25) (2.26) (2.27) The speed given by (2.27) is the maximum possible speed after a braking manoeuvre. However no-slip is very unlikely in such a situation and if the braking mechanism is not fast enough, frictional losses also need to be considered. Again we see that low total mass is desirable as well as a small moment of inertial of the spherical hull. Again, a large moment of inertia of the reaction wheel is favourable. 2.4.2 Advantages and Disadvantages There are several issues arising when using fast spinning ywheels which need to be taken under consideration. 1. Energy consumption is usually unfavourable [5]. A change in rotational speed of the reaction wheel brings about a change in its kinetic energy: ∆Ekin = 1 2 2 Θ(ωnew − ωold ) 2 (2.28) This energy could potentially be recuperated and thereby improving eciency but this of course adds to the complexity of the design. It is likely that most of the energy will be converted into heat when the wheel is decelerated. 2. Gyroscopic eects are unavoidable when the ywheels are spun quickly and are turned around an axis other than their rotational axis. This makes control more dicult. T will cause the T . The power P of the actuator is Θ given by the product of torque and rotational speed P = ω · T . Since the total 3. Velocity saturation will become a problem: A motor torque RW to accelerate as seen before: ω̇ = output power is limited, the motor cannot provide any more torque when the ywheel is spinning too fast. Hence a maximum ywheel speed is reached. Considering these problems, one must design a controller that aims to keep the rate of rotation low whenever possible. Below, reasons for and against using reaction wheels are tabulated: Advantages Energy and momentum can be stored and released. When using braking mechanisms, extremely large torques may be generated (quick transfer of momentum). Still relatively easy to control for 1 DoF as long as gyroscopic eects can be ignored. Can potentially turn in place. Reaction times only limited by motor reaction; torque in desired direction is achieved immediately (no nonminimumphase behaviour). Rather simple for 1 DoF and extendible to several DoFs. Chapter 2. Mechanical Principles 12 Disadvantages May give rise to undesired gyroscopic eects when spun quickly. Velocity saturation will likely be a problem, i.e., torque only available for limited time. Only 1 DoF actuated per RW and hence the mass might become large if several DoFs are to be controlled by RW. For smooth rolling, the CG must coincide with the geometric centre which might be dicult to achieve in practice. With CG at the centre, the system is critically stable. It cannot rest on a slope for innite time. A lot of kinetic energy may become stored in the ywheels, i.e., not all power is used for propulsion. 2.5 Fast rotating ywheels dissipate energy and give rise to ineciencies. Centre of Mass Shift 2.5.1 Modelling Rolling In a study on rolling motion, Armour and Vincent [7] have identied seven distinct design principles that have been used so far in rolling robots. Their study shows that a very popular way to create torques to actuate the outer hull is a shift of the centre of gravity away from the point of contact. A frequently used approach employs a pendulum with two DoFs, hanging from a xed axle inside the spherical shell. It moves by rotating the pendulum around the axis, creating a torque induced by the weight of the shifted mass which is no longer acting through the contact point. Steering is achieved by moving the pendulum around the axis parallel to the direction of motion. Compared to other approaches shown, steering and control of such a design appears simpler, however forward thrust is clearly limited if we only rely on the gravity-induced torque. Another possibility to shift mass is by using a moving cart inside the sphere (like a hamster wheel). In the following a two dimensional dynamic model for a pendulum concept is derived. The basic setup and coordinates can be seen in gure 2.5. The spherical hull (including all components attached to it) has mass M, moment of inertia assumed to be axially symmetric with the CG at its geometric centre motion is captured by the height over ground lowest point). horizontal. The angle ψ h S. IS and is Its vertical (i.e., distance between ground and accounts for rotation and ξ is the position along the In the middle of the sphere a massless pendulum arm of length mounted (idealization). The angle between the vertical and this arm is motor can apply a torque arm at point P, a mass m TS α r is and a between arm and centre of sphere. At the end of this with moment of inertia Ip (relative to its CG) is xed. First we start with the kinematic relations: ψ= ξ , R ψ̇ = ξ˙ , R ψ̈ = ξ¨ , R (No-Slip) (2.29) 13 2.5. Centre of Mass Shift Figure 2.5: Mechanic model of a pendulum based system I rOS = I ṙOS I r̈OS −ξ , R+h −ξ˙ = , ḣ −ξ¨ = , ḧ −ξ − r sin α , R + h − r cos α −ξ˙ − rα̇ cos α = , ḣ + rα̇ sin α −ξ¨ − r(α̈ cos α − α̇2 sin α) = . ḧ + r(α̈ sin α + α̇2 cos α) I rOP = I ṙOP I r̈OP Only if the sphere has ground contact (h = 0), (2.30) the no-slip condition (2.29) applies and normal and tangential forces from the ground adjust themselves such that the kinematic relations are fullled. A rolling friction moment TR is also considered in this case. Now principles of linear and angular momentum can be applied to the sphere, pendulum arm and pendulum mass. Using the fact that the pendulum arm has no mass, it must simply full static equilibrium. Therefore the remaining relevant equations are M ξ¨ M ḧ IS ψ̈ mẍP mÿP Ip α̈ = = = = = = Px + FR −M g − Py + N TS − TR − FR R . Px −mg + Py TS + Px r cos α − Py r sin α (2.31) By eliminating the constraining forces one arrives at the equations of motion: For N need to be eliminated, leaving two equations of motion (2 DoF). Furthermore h = ḣ = ḧ = 0 in this case. Without ground contact N = FR = TR = 0 and only Px and Py need to be eliminated, the case of ground contact, Px , FR , Py and leaving four equations of motion (4 DoF). To allow for losing ground contact, the normal force N has to constantly be moni- tored. Once it gets negative (i.e., the ground would have to hold the sphere back) one has to switch into `ight' mode. The problem with landing on the ground again is that elasticities have to be considered to capture the behaviour correctly. A solution to this problem is deferred for now and will be presented in section 4.4 on page 35. While remaining in ground contact, these second order dierential equations can be decoupled and can be solved numerically. Chapter 2. Mechanical Principles 14 Figure 2.6: Nonminimumphase behaviour of the sphere's horizontal velocity Nonminimumphase Behaviour A further interesting behaviour catches the eye if the graph of the simulation is considered immediately after a torque is applied. In gure 2.6 the horizontal speed ξ˙ has been plotted over time. With initial rest conditions and a step signal on the torque, the sphere at rst moves in a dierent direction than intended. system with such behaviour is called nonminimumphase A which puts limits on its performance (especially reaction times). In this case it is a result of the inertia of the pendulum mass: At rst, this mass has to be accelerated towards the desired movement direction. Due to conservation of momentum, the sphere initially moves in the opposite direction before the gravity-induced torque dominates. Jumping A further very interesting ability of the robot is to jump by moving the centre of mass around. This is relevant for our application since the robot can remain fully in shape (i.e., the spherical hull does not need to deform) and no interfaces are needed. Furthermore it will look rather enthralling to the outside observer if an encapsulated system suddenly manages to loose ground contact and jump over small barriers. From a mechanical perspective, the only external force acting on the sphere that would allow it to jump is the normal reaction force from the ground. Therefore an inner mechanism needs to make sure that this force is larger than the weight for some time such that upward momentum can be accumulated internally. Alternatively one could keep the centre of gravity more or less xed in space but the spherical hull is pulled upwards. Both can be combined and as a rst idea and treatment of the topic we will consider the following strategy: 1. Accelerate some mass upwards on the inside such that the centre of gravity moves up. The force from the ground will exceed total weight during this time and upward momentum is accumulated in the moving mass. Then this mass is stopped relative to the sphere and transfers its momentum to the whole system which will lift o the ground if stopped fast enough. 2. Once in the air, the spherical hull reacts against the inner moving mass and pushes itself further up relative to the centre of mass. This does not change the trajectories of the total CG but the distance between ground and sphere will be increased. The theoretical maximum height that can be achieved with this step only is twice the radius of the sphere, that is, if the hull had no mass at all and was moved instantaneously. 15 2.5. Centre of Mass Shift Figure 2.7: Jumping dynamics Stage 1 As drawn in gure 2.7, the spherical hull (radius cluding the mobile mass is ys , m the one of the mass is R) has mass M, ex- inside of it. The vertical position of the sphere's centre ym . To begin with, the sphere rests on the ground and the mobile mass is at the lowest position possible: ys (0) = R ym (0) = ym0 ẏs (0) = 0 ẏm (0) = 0 . (Initial Conditions) The inner mass is now accelerated upwards by a force F1 > mg (2.32) that is provided by an internal mechanism. Due to the initial rest condition the trajectory is ym (t) = F1 −g m t2 + ym0 . 2 (2.33) ∆t(1) the mass will reach its highest position ym1 in the sphere. For a := (ym1 − ym0 )/2 shall be the distance that the mobile mass can move a At a time clarity, way from the sphere's centre in both directions. The time is given by s (1) ∆t = 2(ym1 − ym0 ) = F1 m −g s 4a . −g F1 m (2.34) At this point in time the mobile mass is stopped relative to the spherical hull and its momentum is shared with the sphere. In an idealized model where the braking happens instantaneously, the momentum remains conserved: − ! + ẏm (∆t(1 ) ) · m = ẏs (∆t(1 ) ) · (M + m) (2.35) Using this initial condition for the trajectory of the sphere, one can nally nd the maximum height (1) hmax (distance between ground and lowest point of sphere) that can be achieved by the rst stage of the mechanism. h(1) max Stage 2 1 = g m M +m 2 F1 − g 2a m (2.36) Once the mass is up, it can be pushed down again so that the spherical hull gains even more height. Again this is trough an internal mechanism and this time F2 > M g (pointing in the opposite direction as F drawn in gure 2.7). Similar Chapter 2. Mechanical Principles 16 to the calculations above, we will consider the internal mass by the force F2 and its weight mg . m being pushed down The initial condition to this process is that the internal mass has no velocity relative to the sphere. Integrating assuming that the sphere remains in the air yields the time ∆t(2) that it takes to bring the internal mass to its lowest position again. s ∆t On the sphere, the same force position at the time ∆t(2) (2) F2 = F2 m 4a + FM2 (2.37) acts upwards and its weight downwards. Its can be computed to h(2) max = F2 −g M F2 m 2a . + FM2 (2.38) Using (2.36) and (2.38) together give the maximum height that can be achieved by such a combined jumping mechanism: 2 2a (2) (1) F1 F2 1 m = + h htot = h max max max g M +m m − g 2a + M − g F2 + F2 m M 2 F F 1 Mm 1 2 = 2a g1 Mm . +m m − g + F2 M − g M +m (2.39) Jumping with a pendulum The above mechanism assumes that a mass can be shifted along a straight line within the sphere. In this section we will explore what changes if a mass on a pendulum arm is used. The principle of exchanging momentum between an internal mass and the outer hull remains the same. The dierence with a pendulum is simply that a rotational motion can be steady (no large breaking or accelerating forces/torques required) and still exchange a large amount of momentum because of the non-linear path. Since the dynamics become rather complicated, we will only consider a sphere resting on the ground and take a look at the normal force that acts between sphere and ground. Using (2.31) with h = ḣ = ḧ = 0 and solving for the normal force N (acting upwards on the sphere, see gure 2.5): N = (M + m)g + mr(α̈ sin α + α̇2 cos α) | {z } | {z } static (2.40) dynamic The static part is only due to the weight of the whole system while the dynamic part has its origins in the swinging motion of the pendulum. If the dynamic part is negative and outweighs the static part, the normal force becomes negative. This is equivalent to saying the sphere would loose ground contact. For simplicity we assume a clockwise rotation of the pendulum (α̇ The dynamic part > 0, gure 2.5). can be further divided into two distinct eects: mrα̇2 cos α centripetal force causes upward force when pendulum is in the upper hemisphere due to the circular path. mrα̈ sin α inertial force causes upward force when accelerating in the right hemisphere and when decelerating in the left hemisphere A possible jumping strategy could therefore be as follows: 17 2.5. Centre of Mass Shift 1. While on the ground, the normal force should be as large as possible to gain upward momentum. Therefore we would like the pendulum to be in the lower hemisphere and decelerating on the right, accelerating on the left. 2. While in thie air, the `normal force' should be made as negative as possible. Therefore we would like the pendulum to be in the upper hemisphere and accelerate on the right, decelerate on the left. Obviously the speed of rotation α̇ has to be chosen in a way such that the pendulum is in the correct position when in air and on the ground and hence allow for a bouncing motion. What will very likely cause diculty is the fact that without ground contact and without no-slip condition, the pendulum reacts against the sphere which begins to rotate, too. Determining the absolute angle of the pendulum arm with the vertical will be challenging. Additionally, in this case it is actually useful to have a spherical hull with a large moment of inertia. This contradicts earlier objectives and a design compromise needs to be established. 2.5.2 Advantages and Disadvantages To sum this section up, reasons for and against using an actuated pendulum are listed: Advantages Easy control as long as the axis remains xed. Constant torque can be achieved until motor power output reached (no misalignment). A 2 DoF pendulum provides one mechanism for both accelerating and steering. Potentially little energy losses (none if at rest) because nearly all energy goes into propulsion. Robot can recover from any position; robust against disturbances . System inherently statically stable since CG below centre of sphere. It can rest in place without energy consumption if the ground is at. Can stay standing on a slope for innite time (although feedback loop required for yaw stability). Extra eects (torques and forces) possible if actuated dynamically (swinging). Disadvantages Usually not omnidirectional and cannot turn in place unless yaw dynamics are used. Maximum torque (in steady-state operation) limited by the mass and the distance over which it can be shifted, i.e., how far the CG can be moved away from the contact point. Maximum speed limited by motor power. The closer the pendulum arm is to the horizontal, the more the roll stability is lost. Relatively large space is required for 2 DoF pendulum that allows for narrow turns. Chapter 2. Mechanical Principles Reaction times might be slow (e.g., from max forward to max backward torque) because mass has to be moved. 18 Sytem is not minimumphase (reacts in wrong direction at rst). Chapter 3 Existing Designs 3.1 Overview The following section aims to provide an overview about prior art of rolling robots. The list is obviously not exhaustive but all relevant concepts are represented by at least one robot. In table 3.1 one can easily see that many dierent designs exist and dierent combinations of propulsion mechanisms have been tried before. The abbreviation N/A means that this propulsion mechanism has not been used. The robots have been ordered in terms of the mechanism used. Robots that use a combination are considered afterwards. Alike to all robots is that they can be full enclosed by their outer shell which protects the components on the inside and reduces the risk of getting stuck since there are no extremities to the shape. Equipping the outer shell with special material that allow the robot to roll over rough surfaces or even swim is possible in theory. Performance overview Table 3.2 provides an overview over the robots presented. Where available, data from literature has been taken, however not all robots had quantitative performance indicators available. 3.1.1 Single Wheel Robot by University of Science and Technology Kraków The rst design is a single wheel robot that uses mass shifting for propulsion and stability in a very interesting way. It has recently been developed by Buratowski and Cie±lak [8]. Their aim was to build a robot that can remain upright for an unlimited time both in motion and when standing. Figure 3.1: Robot developed in Kraków 19 Chapter 3. Existing Designs 20 Figure 3.2: Sphero 2.0 As it can be seen in gure 3.1, inside the big wheel, two independent mechanisms ensure pitch actuation and roll stability: For forward and backward propulsion, a cart can move inside the wheel and shift the centre of mass as required. The balancing mechanism rotates an arc-shaped mass around the roll axis and can therefore be used for steering, too. What is interesting here is that the sideways mass shifting actually occurs at a point above the centre of gravity, similar to an inverted pendulum. This might introduce further agility, however also poses a more dicult problem in terms of roll stability. Unfortunately little further comments have been made on this aspect. In their paper, an important observation is made: The model of the single-wheel robot in the lateral plane is equivalent to a double inverted pendulum whose rst joint is underactuated 1 . Hence control algorithms from the inverted pendulum problem might be useful in this case, too. The authors note that controller that stabilizes the system has been found and proven in practice. 3.1.2 Sphero 2.0 As the name suggests, Sphero is a robot of spherical shape of the size of a baseball and can be wirelessly controlled using smartphone applications. It is a commercial 2 and hence openly available information and data is dicult to Product by Orbotix acquire. However from several reviews [9, 10, 11] on this product and online videos, some properties can be inferred. Although contained in a very durable polycarbonate spherical shell, it is not omnidirectional. When sending a drive command into a certain direction, it often has to reorient rst before it can drive o. This reorientation is however relatively fast 3 and comes with nearly no lateral displacement . Sphero also appears to be able to 4 jump . Unfortunately no information could be found on how this is achieved. Generally Sphero is very agile and it can achieve speeds up to 2 m/s which is already a quick walking pace. The battery lasts about one hour and can be recharged using an inductive charging unit which allows the hull to remain without any gateways. Sphero is also capable of swimming on water, using its rotation to propel itself 1 Buratowski and Cie±lak in [8] p. 104 2 Orbotix, Sphero | Robotic Gaming gosphero.com/ System for iOS and Android, 2013. See http://www. 3 This can for instance be seen in the video Show and Tell: Rollin' With Sphero 2.0 by Tested on https://www.youtube.com/watch?v=078XSj1zKCs 4 Can be viewed in Sphero 2.0 Jump Test by Go Sphero on https://www.youtube.com/watch? v=GQhRN5tTghA 21 3.1. Overview Figure 3.3: Interior of RoBall in a given direction. A nubby cover around the sphere greatly enhances swimming speed. According to the development team, the greatest diculties were fabricating a perfectly round shell which is at the same time lightweight and resistant. They seem to have overcome this issue very well, the shell is even strong enough to withstand an adult person standing on it. Finding durable motors has also been challenging as well as problems with electric discharge that inevitably happens for example when Sphero moves over carpet oors. Internally, Sphero uses the concept of a moving cart with a sprung central member. This can be seen in gure 5 3.2. All relevant components are packed together and contribute to the mass of the robot. This includes for instance the two motors, battery, computation and communication unit and sensors. axis accelerometer and a gyroscope to sense movement. Sphero has a three Two small wheels with rubber tires roll along the inside of the shell and can be controlled independently. The normal contact force is provided by an arm that is extending in the opposite direction and slides (slip bearing) against the inner shell. This way, the wheels never loose contact even in positions where most mass is above then geometric centre of the ball. 3.1.3 RoBall With a diameter of 27 cm, Roball is a relatively large spherical robot. Kabaªa and Wnuk [12] have employed a two DoF pendulum to shift the centre of mass of their robot and hence use a gravity induced driving torque. The motor axle is xed inside the sphere and in gure 3.3 one can see the interior components. Roball is not omnidirectional as its forward or backward motion must be perpendicular to this xed axis in the hull. This means curves and reorientation around the yaw axis can only be performed while in motion. Roball is able to move fully autonomously with all sensing and control being done onboard. Unfortunately little further information on the mechanical aspect has been found. 5 Image by Christopher Harting, taken from [10] Chapter 3. Existing Designs 22 3.1.4 Spherical Rolling Robot by University of Delaware At the University of Delaware, Bhattacharya and Agrawal [13] proposed a spherical rolling robot that uses reaction wheels only. This construction has a spherical aluminium shell, assembled from two halves. Each contains a receiver, motor assembly, rotor and battery. There are two (rigidly connected) rotors on an axis through the geometric centre and one rotor perpendicular to this axis. Special attention was paid to ensure that each component is directly opposite to its pendant in the other hemisphere, such that the centre of gravity is exactly above the contact point for all orientations and the robot does not tip over. The position of the robot is determined by an overhead camera which means not all sensing capability is contained in the robot. The computation is also done externally and signals are sent wirelessly to the motor controls. For modelling purposes, the system was conceived to be an assembly of three rigid bodies, namely the shell and everything xed to it, the double-rotor and the single rotor. Through imposing a no-slip-constraint and formulating angular momentum conservation around the ground contact point, system equations were derived. The inputs for control were the speeds of the rotors and an open-loop control design was implemented, hence no feedback from the motor speeds. However there was, of course, information available about the motion of the sphere by the overhead camera. Minimum time and minimum energy trajectories have been calculated and successfully implemented experimentally, both showing reasonable accuracy. The authors noted an interesting eect with the minimum time specication: One motor was used to provide forward thrust at full power, whilst the other only undertook the steering. For simple inputs simulation agreed well with the actual trajectory however for more complex inputs experiment and numerical simulation results deviated signicantly. The authors suggest the reasons for this to be an unequal mass distribution around the centre of the sphere and the nature of the open-loop control. Clearly, if the robot was to be employed in unknown terrain and outside the laboratory there must be an attitude and position estimation system incorporated since no overhead camera will be available. Although not explicitly stated, it is very likely that the robot cannot follow a straight line forever due to velocity saturation of the driving wheel. 3.1.5 Spherical Mobile Robot by Indian Institute of Technolgy Another interesting concept was put forward by Joshi, Banavar and Hippalgaonkar [14]. Their robot has a truly spherical shape is assembled from two halves. On the inside, each hemispheres has a motor assembly attached to it (including a rotor) and a battery pack with some dead weight. This robot is actuated using two independent internal reaction wheels. The additional weights are necessary to ensure that the centre of gravity is exactly at the geometric centre, such that the sphere is statically stable in all orientations and cannot topple over. The motors are 80 W brushless DC, however no indication of performance of this robot is given. The total mass of the assembly is 3.4 kg which appears relatively heavy when compared to the previously presented models. This might be due to the dead weight, which should ideally be avoided at all since additional inertia is never favourable for an agile robot. The two rotors are placed in a way, that their axes of rotation are perpendicular to each other. Locomotion of this robot is solely based on the principle of conservation of angular momentum. A model of the robot was created by considering generalized coordinates describing the contact point on the rolling plane and the orientation of 23 3.1. Overview Figure 3.4: Robot by the Indian Institute of Technology the sphere. Enforcing a no slip constraint, there remain three DoFs. Control inputs of this robot are the speeds of the two rotors. The authors showed in their paper [14], that this system is fully controllable and can be taken from any given position to any other desired point on the plane with a desired orientation. They further noted that all existing path-planning algorithms cannot be applied and latter must rst be developed. An advantage of this design is its simplicity. It is remarkable, that using only two perpendicular rotors, one obtains a fully controllable system. However it could be the case that this type of locomotion is not very ecient, neither in time nor energy for a given start and end point and it is questionable whether large torques can be generated to make such a device overcome obstacles. Furthermore if one was to realize a similar design, a solution must be found where no additional mass is required, since this will reduce the performance of the robot signicantly. At least, one could replace that mass by additional batteries to lengthen the runtime of this device. 3.1.6 Robot by Massachusetts Institute of Technology In his Bachelor Thesis [15], Schroll aimed for designing a spherical vehicle that overcomes the typical limitations of mass shifting approaches and came up with a mechanism to manipulate angular velocity internally. Schroll identied the space constraint of mass shifts and hence the limited gravity induced torque as a major drawback of this propulsion method. Instead it was proposed to generate changes in angular momentum internally and especially separate generation and usage of angular momentum chronologically. After a few dierent approaches using a single reaction wheel or a single gyroscope, Scholl proposed a design that uses a scissored of gyroscopes together with a pendulum mass shift mechanism. This has the advantage that when the gyroscopes are parallel and spun in opposite directions, they have no net angular momentum and do not introduce undesired gyroscopic eects. A net torque can be generated by tilting the gyroscopes towards each other, so that they both generate a precession torque in the same direction. are schematically drawn in gure 3.5. The gyroscopes This should be relatively energy ecient because the kinetic energy of the gyroscopes remains constant and only the angular momentum is used. There are however several problems associated with such a concept: Chapter 3. Existing Designs 24 Figure 3.5: Robot by MIT Figure 3.6: Gyrover Once the ywheels are not parallel anymore, their angular momenta do not cancel out and inuence the dynamic behaviour. If they are misaligned, they cannot supply any more torque and have to be brought back to their initial orientations. This may cause undesired torques of the ywheels keep spinning or cause large energy losses if ywheels are stopped. Furthermore large stresses on the structure are to be expected which needs to be built accordingly. 3.1.7 Gyrover Gyrover is a single wheel rolling robot that uses a gyroscope to stabilize its motion and orientation. It was developed by Brown and Xu [16, 17, 18] at Carnegie Mellon University. This robot is not truly spherical but has an ellipsoid shape and Brown and Xu opine that this shape is in fact more suitable since it exhibits natural steering behaviour. Gyroscopic eects are used for steering and stability, but not for propulsion. Figure 3.6 shows a picture and the internal components of Gyrover. At the heart of Gyrover, there is a ywheel spinning with its axis parallel to the roll axis in forward direction. This leads to self-correcting behaviour due to the eects of precession: Any deviation from the upright position (i.e., a lean to one 25 3.1. Overview side) will generate a torque on the gyroscope due to gravitational forces, making the gyroscope precess around the yaw axis. This is a steering action, engendering a curved path to be followed. Due to the centripetal force provided by the ground through friction, there is a righting moment that will bring Gyrover back towards its upright position. Since the angular momentum of the ywheel is not coupled with the rotational speed of the outer surface, this eect is also present when the robot is at rest. However due to the lack of a centripetal force (since there is no motion), the robot will continue to precess around the yaw axis (i.e., turn in place) and needs an inner mechanism such as shifting the CG if it was to achieve lateral stability again without starting to move. Active steering is achieved by a mechanism that allows the axis of the internal ywheel to be tilted manually around the roll axis. If for example an actuator tries to tilt the ywheel to the left, the outer hull leans entirely to the same side whilst the ywheel remains roughly aligned, because it serves as an internal reference for attitude due to its high mass and angular momentum. As the CG of Gyrover is now left of the contact point, a torque by gravity trying to make it lean even more to the left is experienced. This however is prevented by the gyroscope which starts to precess leading to the correcting eect as described above. Moving the gyroscope back into its nominal position nishes the manoeuvre. Thrust is produced by means of a hanging pendulum suspended at a central axis, xed to the outer hull. A separate drive motor tries to shift the pendulum into the desired direction (forward or backward), causing a gravity induced torque that makes the device roll. There are three working prototypes of Gyrover. The rst has a diameter of 29 cm and a mass of 2.0 kg. It shows good high-speed performance (10 km/h) even on rough terrain, moved through a gravel pile and can also stand in place. The thrust produced by the pendulum is approximately limited to 25 percent of the vehicle weight which means it can climb slopes with a 25 % incline without dynamic eects. Although the robot is not spherical and can therefore fall on its side, it has the ability to right itself from the rest (side) position with the help of the tiltable ywheel. Brown and Xu found out, that as forward speed increases, the tilting torque of the gyroscope required for steering increases strongly, because the angular momentum of both the wheel and the ywheel exhibit self-correcting properties. The authors have identied a design trade-o between a high angular momentum suitable for static stability and at low speeds and a lower angular momentum to allow steering at higher speeds. Therefore a ywheel with adjustable moment of inertia and rotational speed would be advisable. The high energy consumption by the motor driving the ywheel has been addressed in the second prototype by placing it in an evacuated shell. This has cut down power consumption by 80%. 3.1.8 Gyrobot Gyrobot [19, 20] is very similar to Gyrover (see 3.1.7) except Al Mamun made some modications to the actuation mechanism. Eectively the pendulum mass-shift mechanism was combined with the gyroscope. Gyrobot is a single wheel vehicle where a gimbaled gyroscope disc is hung from a central axis. This disc is made of carbon steel and rotates at 10 000 rpm. a large angular momentum stabilizes the wheel in an upright position. Such Shifting the disk forward moves the centre of gravity in front of the contact point and the device starts to move. When a tilting torque in the roll direction is applied to the gyroscope, it begins to precess. Both the eect on the roll movement and the precession are exploited for steering. Figure 3.7 shows how the interior of Gyrobot looks like. The ywheel is smaller than the one from Gyrover, however the CG is also likely to be lower. Chapter 3. Existing Designs 26 Figure 3.7: Gyrobot 3.1.9 Reactobot A concept that uses both a reaction wheel and mass shifting by means of a pendulum is Reacobot, designed by Biswas, Bhartendu, Kadam and Seth [21, 22]. It is a single wheel robot that has an actuated pendulum arm suspended from the axis of revolution of the wheel. It is fully contained in a hollow disc. At the end of this arm, there is an actuated reaction wheel mounted with its axis parallel to the roll axis of the robot. Figure 3.8 shows all parts schematically. For forward and backward motion, the pendulum arm is pitched forwards or backwards respectively, and the heavy reaction wheel at its end causes the centre of gravity of the whole system to shift. Due to geometry and mass distribution, the robot is inherently stable along the pitch axis but unstable in roll. Therefore the reaction wheel is responsible for roll stability as well as steering. Curves can be accomplished by tilting the robot to one side while in motion. Kadam and Seth [22] designed an LQR controller for the system that ensures roll stability and brings the reaction wheel to rest at the same time. Using this controller, given trajectories can be followed. It is crucial that the reaction wheel is at low speeds most of the time for three main reasons: 1. Avoiding gyroscopic eects, 2. energy consumption and 3. roll stability must be achieved before the power output limit is reached. The latter becomes especially important because the robot cannot recover once it has fallen onto its side. This means it has to be set up to start rolling and any external disturbance that might cause the robot to tip over will bring about a system failure. One would expect that the faster the wheel is moving, the more stable the roll axis is anyway, because the wheel exhibits gyroscopic properties, too. A problem with this design is however, that while accelerating, the RW axis is not parallel to the ground (pendulum lifted up) and therefore eects the yaw motion as well. Hence several DoFs become coupled and pose a diculty in controlling. On the positive side, this setup manages to follow given trajectories using only two actuators. 27 3.1. Figure 3.8: Reactobot Figure 3.9: Robot by Kobe University Overview Chapter 3. Existing Designs 28 3.1.10 Robot by Kobe University A spherical rolling robot that uses one ywheel only for propulsion in a very elegant way has been put forward by Urakubo, Osawa, Tamaki et al. from Kobe University [23]. The centre of gravity of the entire system coincides with the geometric centre. The system is composed of three main bodies: a gyroscope, a gyroscope case (inner hull) and the outer shell. The most interesting part of this concept is that it uses two concentric spherical hulls. Inside the inner hull, a gyroscope spinning at a large angular velocity is rigidly mounted and hence this inner hull's attitude is stabilized by the gyroscope. The connection between the hulls is realized through four rubber rollers, two of which are actuated. These rollers can be driven by two independent motors in directions perpendicular to each other and the gyroscope axis. This allows the outer hull to be rotated relative to the inner hull. With this mechanism, the angular momentum from the gyroscope can eectively be transferred to the outer hull and therefore provides a torque for movement of the entire system. An important condition is that the rollers do not slip sideways which poses high requirements on the mechanical design. With another motor that directly drives the gyroscope and reacts against the inner hull, there are three independent torques that can be applied to the system. According to the considerations under 2.1 (page 3), all available DoFs should be controllable. The system equations are derived by applying an overall conservation law of angular momentum over all three bodies (spinning disc and both hulls). Through controlling all three input torques rolling has been achieved. One might expect nutation to be a problem, however this was not detected experimentally. The authors suggested that this might be strongly damped and does therefore not show. In experiment, the robot with a suitable controller was able to track a given velocity prole with a small error. A problem that was already identied by the developers was that rolling friction will change the total angular momentum and should therefore not be neglected in the model, which they did in their rst approach. 3.2 Conclusions from Existing Designs From the above designs one can see that nearly all propulsion mechanisms can be combined in dierent forms and there are still many more possibilities to try out. It also became clear that two motors are in fact enough to adequately control a spherical or single wheel robot. The most used concept was pendulum mass shift and interestingly this goes well together with reaction wheels and gyroscopes. A possible reason for the frequent use is the relative simple control design due to oneaxis actuation only and the ability that it can provide a constant torque without issues of misalignment or the like. Very often is has been noted that these systems are relatively dicult to control and previous algorithms for path planning usually do not apply. Therefore it is very likely when proposing a new design that one cannot resort to existing controller software but must develop the latter specically for this robot. In almost all cases, the design goals are to achieve maximum torque for acceleration whilst keeping mass of the vehicle within reasonable limits and permit good steering capability [2]. Furthermore the robot should be able to overcome obstacles and ideally also climb stairs. Concerning size, one must consider that enough space needs to be available for ywheels, other propulsion mechanisms including motors and all electronics, for example micro-controllers and battery packs. Furthermore, the larger the radius, the easier it will be to overcome obstacles and the lesser chance to get stuck in a hole. However a large hull will have a relatively high 29 3.2. Conclusions from Existing Designs moment of inertia and prohibits fast acceleration, whilst at the same time being susceptible for external inuences, namely wind. Furthermore we might want the robot to explore rather small cavities or rest on small platforms, therefore a good compromise between the above arguments must be found. Chapter 3. Existing Designs 30 Table 3.1: Propulsion mechanisms of existing robots Robot Shifting Mass Kraków Mass moves like cart for pitch mass torques, can shift Reaction Wheel(s) Gyroscope(s) N/A N/A N/A N/A N/A N/A Two RW mounted perpen- N/A another laterally for roll stabilization and steering (roll) Sphero Mass can move like a cart inside sphere to control pitch and yaw, roll is inherently stable (passive). Sprung member ensures permanent contact RoBall Suspended mass (2 DoF pendulum) for forward and backward acceleration and sideways steering Delaware N/A dicularly control roll and pitch Indian Inst. N/A Two RW mounted perpen- of Technology MIT Robot pitch Suspended mass (1 DoF pendulum) for N/A Scissored Suspended mass (1 DoF pendulum) for pendulum) for Gyroscope used for steerity N/A Gyroscope stability forward and backward acceleration Reactobot University RW with axis in drive di- pendulum)for forward and rection used for roll stabi- backward acceleration lization and steering (roll) of Kobe used and for N/A A ywheel can be acceler- Two ated and decelerated rotate the gyroscope rela- independent rollers tive to the outer shell Table 3.2: Performance of existing designs Robot Diameter [cm] Mass [kg] Speed [m/s] Sphero 7.5 27 60 29 43 20.6 0.180 1.42 3.4 2.0 3.24 4.1 2.0 1.9 RoBall Indian Inst. of Techn. Gyrover Gyrobot University of Kobe roll steering (yaw) Suspended mass (1 DoF N/A gyro- ing (yaw) and roll stabil- forward Suspended mass (1 DoF of ing torque (pitch) N/A and backward acceleration Gyrobot pair scopes to supplement driv- forward and backward acceleration Gyrover N/A dicularly control roll and 2.78 0.49 Chapter 4 New Design Ideas Having seen a variety of dierent concepts that have already been designed and build, we would like to improve on existing designs and propose some new ideas. The rst step of this process would be to introduce the requirements that we want our robot to full. 4.1 Requirements 1. Basic abilities [a] Move along any given trajectory on at ground for an innite time (only limited by battery power) [b] Be able to move up slopes at constant speed for innite time [c] There should be no dead spot on the spherical hull where the robot cannot recover from [d] Can stand and rest at a given position on a slope for innite time without using excessive energy 2. Advanced abilities [a] Make use of dynamic eects to climb steeper slopes and longer distances and recover from cavities [b] Should be able to turn in place in order to move o in any direction and make very sharp turns. (Requirement obsolete if the robot is inherently omnidirectional) [c] Capable of rapid acceleration, i.e., be very quick in responding to direction and speed changes. It should look very agile from the eyes of an observer. [d] Being able to overcome obstacles by jumping over them. 4.1.1 Conclusions Very long straight-line movement can hardly be achieved by a RW or Gyroscope only. These will eventually get velocity saturated and/or misaligned so that no further torque can be provided. Therefore some form of shifting mass mechanism is necessary that allows for a constant gravity-induced torque. Taking another look at the review on rolling motion [7], car-driven (possibly in connection with sprung central axis) and pendulum are the only feasible mechanisms to work inside a constrained space. Let us have a closer look at those: 31 Chapter 4. New Design Ideas Car driven 32 If the car is suciently at, this appears to be a very space ecient solution. The mobile cart must obviously be secured against falling over. A cage or any other constraint that keeps its wheels on the inner spherical surface is acceptable. The energy supply to the car appears to be critical. The mass should be as far away from the centre of the sphere as possible to allow for maximum gravity induced torque. Depending on the design of the car, it may be omnidirectional (car must be able to go in both directions on the surface equally). The reaction time might not be ideal because the cart has to move to the opposite side of the sphere in the extreme case. Pendulum A pendulum would require much more space to move around, especially if it is has two DoF. The mass distribution is not as good as with a car driven system but in this case the mass itself could be used for another purpose, for instance it could be a spinning wheel that exhibits gyroscopic properties. 4.1.2 Fullling the Desired Requirements Basic abilities [a+b] The ability to climb slopes or roll on at ground with friction for innite time is equivalent to the being able to provide a constant torque in the same direction. This would be automatically fullled by both mechanisms presented above. The slope incline is however strictly limited to a certain value depending on how far the CG can be shifted (see section 4.4 for maximum slope). [c] Neither for car driven nor for a pendulum model there are any dead spots. Both systems are furthermore inherently stable because their centre of mass dose not coincide with the geometric centre. [d] Similarly to the rst condition, standing on a slope for an innite time requires a constant torque to react against the incline. A reaction wheel or a gyroscope will eventually become velocity saturated respectively misaligned which shows once again that a mass-shifting mechanism is necessary. It must be noted that not only roll must be controlled but also yaw (which will be unstable). Advanced abilities [a] Any device can use its translational and rotational momentum to climb up steeper slopes (i.e., gather momentum beforehand). Furthermore, both shifting mass mechanisms can perform a swinging-motion. This leads to non-constant torque which is likely to be stronger than the static torques at its peaks, but whether this is the case on average must yet be determined. A further reaction wheel could be accelerated in opposite direction of rolling motion, thereby creating a time-limited torque to move the sphere further up. If the reaction wheel was turning in direction of spin of the sphere, its momentum could be utilized directly by synchronising the this wheel with the sphere. Finally, gyroscopes can induce very large torques supplementing the static shifting mass torque for some time. The reaction torque by the driving motor will however cause over movements that need to be counteracted upon. [b] Truly turning on spot is only possible if internally the angular momentum in the direction of the yaw axis is changed. This would be most easily achieved with one reaction wheel aligned with this vertical axis or several reaction wheels whose combined momentum is changed along this axis only. However one must also consider that with an omnidirectional car, turning in place is not necessary. 33 4.2. Omnidirectional Car-based Figure 4.1: Design Idea: Cart and Gyroscope [c] This is again closely linked with the rst point and calls for a mechanism that provides extra torque in the drive or lateral direction when a control input change happens (i.e., for quick responses). [d] Jumping is a complex manoeuvre that requires either shifting the internal centre of gravity in a vertial direction or being able to 'push away' from the ground, or both (see section 2.5.1). A pendulum is more likely to achieve this, but any design could be equipped with a suitable mechanism. 4.2 Omnidirectional Car-based One possibility to achieve a highly agile yet powerful actuation for a sphere would be to combine a car-based mass shifting mechanism with a spinning disc used as reaction wheel and gyroscope. Due to the high mass of a ywheel, one would use only a single one mounted in the centre of the spherical hull and actuated in all three rotational axis (3 DoF). The car must be able to go in any direction and would make the robot fully omnidirectional. For quick manoeuvres or extra propulsion the ywheel would in turn provide extra torque for a limited time. A potential design has been sketched in gure 4.1 To the inside of the spherical hull at two opposite points, rods are xed. On one side, this rod has a rotational joint along the axis of the rod, on the other side there is a motor. This allows a ring-track to be mounted such that it can turn around a xed axis relative to the outer hull. On this ring track, an actuated cart is attached that can move along the entire circumference of this round track. The ring track must be attached to the rods in a way that allows the cart passing through these points. One possibility to do so has roughly been sketched. Further inside the sphere a ywheel is mounted using two gimbals, both of which are actuated. The spinning speed of ywheel itself can be controlled by a further motor. This setup would require ve actuators which is two more than DoFs the sphere has (no-slip condition). One motor would need to be on the car itself to make it go aground the ring track. The energy supply could be through an electrical potential dierence between the two tracks of the ring which is picked up by the car. This Chapter 4. New Design Ideas 34 would however require sliding electrical contacts. Furthermore the energy supply for the other motors is another dicult issue due to many moving parts and joints; a separate battery for each actuator might be the only viable solution. Finally this design has a high degree of mechanical complexity and would probably be dicult to build and maintain. In the following, advantages and disadvantages of such a design are listed: Advantages A thin trolley on the ring track leaves most of the space inside the sphere for other components, i.e., very space ecient. The centre of gravity is relatively high since the ywheel is mounted at the centre. This allows the robot to overcome small steps relatively easily with some inertia. Concept is omnidirectional. It can drive o in any direction without turning beforehand. Gyroscopic eects can be completely 'turned o ' when not needed because the rotational motion of the gyroscope is decoupled from the one of the sphere. Gyroscope may be used to actively stabilize the motion and reject disturbances, even when no extra torque for a specic manoeuvre is required. When the trolley and the gyroscope are used together, a gravity induced torque can be amplied to go around curves without much eort. Disadvantages A high CG is disadvantageous when climbing slopes. Five actuators are three more than really necessary and the energy supply of the inner ones will likely pose diculties. The ywheel might be misaligned to give the desired torque at any time. Furthermore a strong outside disturbance might lead to nutation and energy and intelligent control is required to damp such undesired motion. The trolley has a dead spot if located on the axis of its ring track. If it wants to go sideways at this point, the track has to react very quickly which creates an undesired torque on the outer hull. 4.3 Complicated Pendulum with Flywheel Another idea is to use a two DoF pendulum hung from a xed axis in the sphere with the mass at the end being a ywheel that can be twisted and accelerated. This idea is shown in gure 4.2. This approach would use four actuators and dierent operation modes are possible. For instance, the ywheel could be positioned with its axis in the direction of motion and therefore only used for roll stabilization whereas the pendulum does all the rest. Further highly dynamic motions that exploit the gyroscopic properties are imaginable. listed below: Advantages Reasons for and against such a design are 35 4.4. Simple Pendulum Figure 4.2: Design Idea: Pendulum with Gyroscope Requires only four actuators CG is restively low so good slope climbing ability can be expected and larger centre of gravity shifts are possible than with the trolley approach Disadvantages Most torques by gyroscope act through the central actuator that connects the upper pendulum arm with the axis in the sphere. This motor must therefore be able to produce (withstand) extremely high torques This design is not fully omnidirectional, however it can easily employ the gyroscope to align the pendulum for the desired forward motion direction. Nearly all space inside must be reserved for the ywheel to move in any possible position 4.4 Simple Pendulum Usually, a suspended mass used as pendulum is considered to have a maximum torque when it is held horizontally, because then the shift of centre of mass is greatest. For example the developers of GyroBot mention that acceleration torque T is limited by the suspended mass [19] : T = mg · r · sin(α) (4.1) This is reasonable if we want a steady-state behaviour, i.e., keeping this torque for a longer period of time. However we propose a new design in which this constraint is being ignored. For the shell, the torque that is seen is exactly the motor torque that acts on the pendulum arm and the central axis of the sphere. Nothing prevents this motor to exceed the maximum torque by gravity. Obviously the pendulum arm may lift further than 90◦ and some very interesting behaviour can be observed. The basic equations that should capture these dynamics have been developed in 2.5 (page 12). However they do not accurately reect the jumping dynamics because the Chapter 4. New Design Ideas 36 switching between being airborne and ground contact is too strict and no elasticities are considered. Especially the latter are very much relevant when it comes to bouncing. As a dierent approach, a model of the sphere and a point mass at the end of the pendulum has been implemented in a multi-body simulation program called SimMechanics x ϕ gives the angular orientation. by MATLAB/Simulink. The model is two-dimensional with the direction pointing along the ground, y upwards and By using the signum function −1 sign(x) := 0 1 if if if x < 0, x = 0, x > 0, (4.2) dierent aspects have been modelled as follows: Ground Normal Force: The normal force exerted by the ground on the sphere is modelled as a one-sided linear elastic spring: N = k1 (|y| − y) 1 2 (4.3) This formulation gives zero force when there is no ground contact (y for contact (y ≤ 0) > 0) and a force proportional to ground penetration. Ground Damping: To avoid perfectly (and unrealistic) elastic behaviour, a damping force is added to the normal ground force from above: Nd = −d1 ẏ sign(|y| − y) No-Slip: (4.4) The ground tangential force should prohibit any slipping while the sphere has surface contact. This is obviously not very realistic because in reality, slipping is well possible and the friction force actually decreases once traction is lost. In this case, the friction force is proportional to the relative motion between ground and contact point of the sphere: FR = −k2 sign(|y| − y)(ẋ + Rϕ̇) Friction: (4.5) All friction eects such as air resistance and other non-ideal be- haviour is accounted for in a linear damper term that opposes any motion: Ffriction = k3 −ẋ −ẏ (4.6) Under section B, gure B.1 a SimMechanics model has been built using the above force laws to simulate the ground. Performance This part is concerned with the performance of a simple pendulum approach. With the conditions described in this section there are several indicators to consider: The maximum driving torque that can be maintained without dynamic eects (i.e., only the gravity induced torque) is gravity Tmax = mgr (4.7) 37 4.4. Simple Pendulum assuming that the maximum motor torque exceeds this value. Considering a countertorque due to slopes with angle of elevation γ T slope = (m + M )gR sin γ (4.8) we can use (4.7) and (4.8) to calculate the maximum slope that can theoretically (no-slip) be climbed using such a steady-state (α̇ ! gravity Tmax = T slope ⇒ γmax = 0) arrangement: mr = arcsin (M + m)R (4.9) If friction is still large enough, obviously steeper slopes can be climbed for a limited amount of time using momentum and stronger motor torques. The acceleration on at and frictionless ground that can be achieved by an arbitrary motor torque T drive that acts on the sphere and the pendulum can be taken from (2.23) to be ẍ = T drive R(m + M ) + ΘS R . A good approximation for the moment of inertia of a spherical hull is and hence ẍ = T drive . R(m + 35 M ) (4.10) ΘS = 23 M R2 (4.11) Chapter 4. New Design Ideas 38 Chapter 5 Conclusion This research project has shown that there are several options available to propel a round robot. Among the three main categories gyroscopes, reaction wheels and mass shifting the latter appears to be predominant for most applications, mainly because it can provide constant torque which does not get exhausted. Previously many dierent round robots have been built and most use more than one propulsion mechanism. Especially combinations using mass shifting together with one out of the other two categories appears to be popular because it adds extra stabilization and torque capability to the vehicle. Some very elegant combinations are possible and there are still numerous designs which are yet to be tried out. Several robots presented are not spherical but rather ellipsoid like a single wheel and arguably have better dynamic properties. However the inherent roll stability is lost and an extra mechanisms is required to retrieve stability. Encapsulated systems in general provide good protection to the (fragile) interior parts and hence show promise to be more durable especially when operating in a rough environment. The manufacturing of such a sealed hull as well as energy supply remains a challenge and must be thoroughly considered when designing a new robot . So far, only Sphero is capable of being charged up without opening the hull, that is by induction charging. The mechanical analysis as well as simulation models could be used in further works to elaborate on a new concept for rolling robots. A rst idea is presented in the following section. The ndings of previous works showed that it is crucial to fully develop a concept into a prototype because one can hardly identify the full dynamic behaviour without experiments. As controller design seems to be a major hurdle, a good theoretical model and parameter identication using the prototype are necessary to capture the system adequately. 5.1 Outlook As starting point for further works, one might want to consider a two dimensional model to explore the simple pendulum concept from 4.4. For instance a single wheel robot with a bike wheel could be worth a try: Typical bike wheel data 1 of a standard rim 700 wheel (29 inch diameter) with 32 spokes and including tire, tube and axle: Polar moment of inertia Mass Ip m = = 0.0885 kgm2 1.264 kg (5.1) 1 Taken from: T. Compton. (1998). Analytic Cycling - Performance and Wheels Concepts [Online, Accessed 2013/12/30]. Available: http://www.analyticcycling.com/WheelsConcept_Disc. html 39 Chapter 5. Conclusion 40 Figure 5.1: Possible System using a bike wheel with test stand (side and front view) The objectives for the prototype are Fast acceleration Being able to hop/jump dynamically Can jump with only little horizontal displacement (i.e., can stay in one place while jumping) Considering the dynamics of jumping from section 2.5.1 (p. 14) there obviously needs to be a compromise between large wheel moment of inertia (prevents wheel from rolling while jumping) and small mass (faster acceleration, higher jumps). The basic setup of such a bike wheel setup could be as follows: A lightweight but rigid pendulum arm is attached to the axle of a bicycle wheel through a joint that permits free rotation around an axis collinear with the wheel axis. A motor could be xed onto this arm with its rotor rigidly connected to the wheel axle so that it can exert a torque between pendulum arm and bike wheel. All batteries and computation units will be mounted on the arm, too. A sensor measuring the angular velocity between wheel and arm is required and one also needs at least one inertial measurement unit on both wheel and pendulum. Since a bike wheel is not stable in the roll direction, a test stand needs to be build which removes this DoF. Figure 5.1 shows how this may look like. What makes this concept dierent from previous ones is that is uses a pendulum mass-shift mechanism to perform highly dynamic tasks by using it as an excentric ywheel. This has never been tried out before. Furthermore there are some advantages: Relatively simple design compared to manufacturing a sphere - uses available parts such as bike wheel. Should be able to jump which no robot but Sphero achieved so far. 41 5.1. Outlook Can operate in both a steady-state and dynamic mode. An extension to the general 3D is case possible. Relatively lightweight since no ywheels or gyroscopes are necessary (and no dead weight). No undesired gyroscopic eects and no energy losses due to keeping ywheels spinning at high rotational speeds. Chapter 5. Conclusion 42 Appendix A Gyroscope Dynamics In this section the equations of motion of a gyroscope are derived. As done in previous parts of this work, only a spinning disc with constant density will be considered. The centre of gravity of this disk shall rest in space and due to symmetry, the moment of inertia tensor can be simplied in the body-xed coordinate system K ΘS = diag(ΘA , ΘB , ΘB ) K: (A.1) Through three elementary rotations (XY Z ) an arbitrary orientation of the bodyK -system can be achieved: The inertial system I is xed in space. By rotating around a common x-Axis by an angle ψ we reach the B -system. The following C system is rotated around the y -axis by an angle α. Finally the K -system is obtained by rotating around the z -axis by an angle β . These are all elementary rotations xed where the transformation matrices and rotational velocities can be found straight away. For notation, please refer to the symbols section (page vii). The coordinate transforms are schematically drawn in gure A.1. 1 0 0 sin ψ = 0 cos ψ 0 − sin ψ cos ψ cos α 0 − sin α 1 0 = 0 sin α 0 cos α cos β sin β 0 = − sin β cos β 0 0 0 1 ABI ACB AKC ψ̇ 0 I ω IB = B ω IB = 0 0 α̇ B ω BC = C ω BC = 0 0 0 C ω CK = K ω CK = β̇ Figure A.1: Gyroscope with coordinate system transform 43 (A.2) (A.3) (A.4) Appendix A. Gyroscope Dynamics 44 We further dene the absolute angular velocity components of the disk in the bodyxed coordinate system to be ωx ωy . K Ω := ωz (A.5) Now we are ready to compute the angular momentum of the disk with respect to its resting CG. The derivative has been calculated using the Euler-derivative rule for rotating frames of references: K LS K (L̇S ) ωx ΘA = K ΘS K Ω = ωy ΘB , ωz ΘB (A.6) ω̇x ΘA = K L̇S + K Ω × K LS = ω̇y ΘB + (ΘA − ΘB )ωx ωz . ω̇z ΘB + (ΘB − ΘA )ωx ωy (A.7) This result (A.7) is exactly Eulers equation for a gyroscope [24]. The next step is to express the angular velocity vector K Ω in terms of the angles For the sake of (and their derivatives) used in our coordinate transformations. clarity, some steps are omitted in the following calculation. ωx ωy = K Ω ωz := K ω IK = AKC ACB B ω IB + AKC C ω BC + K ω CK ψ̇ cos α cos β + α̇ sin β . . . = −ψ̇ cos α sin β + α̇ cos β . ψ̇ sin β + β̇ = = K ω IB + K ω BC + K ω CK (A.8) Now we can nally use the principle of angular momentum. This will be evaluated in the body xed-coordinate system using (A.7) and (A.8). In order to shorten the expression cosine and sine functions are abbreviated by = c and s resprectively. MS = K (L̇S ) (A.9) K ΘA ψ̈ cα cβ − ψ̇ α̇ sα cβ + β̇ cα sβ + α̈ sβ + α̇β̇ cβ ΘB −ψ̈ cα sβ + ψ̇ α̇ sα sβ − β̇ cα cβ + α̈ cβ − α̇β̇ sβ ΘB ψ̈ sβ + ψ̇ β̇ cβ + β̈ + (ΘB − ΘA ) ψ̇ cα cβ + α̇ sβ −ψ̇ cα sβ + α̇ cβ This is the nal equation to simulate the behaviour of the gyroscope in the presence of dierent external torques. If the external moments are given as a function of time, this system can be solved numerically. Singularity One clear disadvantage of using Euler-rotations in describing the orientation of a rigid body becomes evident here. have the same inuence on the orientation while the equations show a and β anymore (they α remains in this position). This is singularity because one cannot tell the dierence known as α = 90◦ between ψ For the case of gimbal lock 1 . Using quaternion description might have been useful here. 1 Gimbal lock is a situation where two axes of rotation become aligned. 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