From Kinematics to Arm Control
a) Calibrating the Kinematics Model
b) Arm Motion Selection
c) Motor Torque Calculations for a Planetary
Robot Arm
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a) Calibrating the Kinematics Model
• Resolution, Repeatability and Accuracy
• Causes of Mechanical Error
• Calibrating the kinematics model
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Resolution, Repeatability and Accuracy
1. Resolution: smallest incremental move that
a robot can physically produce.
2. Repeatability: a measure of the robot’s
ability to move back to the same position
and orientation over and over again. Also
referred to as ‘precision’.
3. Accuracy: the ability of the robot to move
precisely to a desired position in 3D space
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Repeatability (Precision) and Accuracy
Poor Accuracy
Poor Repeatability
(Assuming Gaussian
Distribution)
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Poor Accuracy
Good Repeatability
Good Accuracy
Poor Repeatability
Good Accuracy
Good Repeatability
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Causes of Mechanical Error
Denavit-Hartenberg (D-H) Parameters:
Link Length (d); Link Offset (a);
Link Twist (α); Link Rotation (θ)
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Calibrating the Beagle 2 Arm
Typical Vicon measurement setup
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Beagle 2 FM Arm with
Vicon Markers
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Oxford Metrics Vicon System
‘Object’definition in Vicon system
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Calibrating the ARM Kinematics Model
The simulated (perfect) arm is moved to a
number of positions using the arm IK model.
The end position for each simulated Vicon
marker is measured. The real arm is then
commanded to the same number of predefined
positions in Cartesian space using the joint
values calculated from the arm IK model. The
actual end position for each real Vicon marker
is measured. The real and simulated Vicon
marker positions are compared and joint
offsets (corrections) are calculated for the
simulated arm’s IK model using a least-squares
fitting method.
The result is that the simulated arm with the
corrected IK model produces identical
With
positioning results as the real arm. i.e. The
IK Model
simulated arm behaves as per the real arm.
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IK model corrections
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b) Arm Motion Selection
• Joint-by-Joint Motion (JJM)
• Slew Motion (SM)
• Joint Interpolated Motion (JIM)
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Joint-by-Joint Motion (JJM or JBJ)
• Strengths: Simplest of the possible motion
types; good for un-stowing and stowing an arm
as the major joints can be moved one after
another. Typically no straight-line trajectory
inverse kinematics solution will exist for this
type activity anyway. Good at minimising power
usage per unit time.
• Weaknesses: Joints cannot move
simultaneously, hence cannot be used for
straight line (continuous) trajectories. Rather
payload motion will be in an series of arcs.
Large total time to complete all joint motions.
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Slew Motion (SM)
• Strengths: Also known as “Simultaneous Full
Speed” (SFS) motion. All joints that require
motion start simultaneously at default joint
speeds (usually maximum speed). Quicker joint
motion completion when compared to JJM. SM
can be used for payload straight-line trajectories.
• Weaknesses: Greater power usage per unit
time compared to JJM. Joints with smaller
angular movement complete their motions
before those joints with larger angular
movements. Smooth straight-line trajectories not
possible. Payload motion tends to zig-zag about
the desired straight-line trajectory.
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Joint Interpolated Motion (JIM)
• Strengths: All joints that require movement
start simultaneously and stop simultaneously.
Robot attempts to achieve simultaneous start and
stop by setting the speed of each joint angular
motion to be proportional to the angular distance
to be travelled for each joint. Produces smooth
straight-line trajectory motion.
• Weaknesses: Greater power usage per unit
time compared to JJM. Greater computational
overheads when compared to SM.
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Knot Points
•
•
Straight-line trajectory requires knot points
(from spline mathematics) to be generated
between the start and finish positions.
The desired straight-line between the payload
start and finish positions can be uniformly
divided into intermediate points in Cartesian
space (called knot points)1. Each of these
points can be put through the IK model to
generate the joint angles required to move the
arm to each of these intermediate points in
turn; whilst setting each joint speed (e.g.
proportional for JIM) to the required angular
motion.
1 Taylor, R. H., Planning and Execution of Straight Line Manipulator
Trajectories, IBM J. RES. DEVELOP, VOL. 23, NO. 4, pp. 424-436,
JULY 1979.
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Start
position
Knot
point
Finish
position
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c) Motor Torque Calculations for a
Planetary Robot Arm
• What is torque, and why do we need it?
– Torque (τ) is defined as a turning (or twisting)
force
– A robot arm joint motor, with an attached
linkage and payload, has to generate a torque to
be able to move the linkage and payload
– The magnitude of τ depends upon the payload
force, and the length of the joint motor linkage
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Torque Equation - 1
τ=F×l
where τ is the magnitude of the torque (N·m),
l (in meters) is the linkage length, and F (in Newtons)
is the magnitude of the force.
In the vertical plane, the force acting upon an object (causing
it to fall) is the acceleration due to gravity (for Earth, g ≈ 9.81 m/sec2).
Therefore F = m × g, where m is the magnitude of the mass of the
object (in kg), and F is normally referred to as the object’s weight.
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Torque Equation - 2
sin 90° = 1
sin 45° = 0.7
sin 0° = 0
τ = F × l × sinθ
F
F
L
Linkage
(length l)
Joint
Motor
F
θ = 45⁰
θ = 90⁰
L
τ is maximum
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θ = 0⁰
Pivot
Point
L
maximum > τ > zero
L
τ is zero
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Real Arm Linkage Torque
Joint
Motor
F
L
Pivot
Point
Weight
Motor
Weight
Linkage
WM
WL
L
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(Use maximum (holding) torque case)
Real motors and linkages
have mass and therefore
weight under gravity
Assume linkage mass is evenly
distributed and the weight is located
at the centre of the linkage length
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Arm Joint Motor Torque Calculations
WP
WM1
WL1
WM2
WL2
WL3
Payload
M1
L1
M3
M2
L2
L3
τM1= (WP × L1) + (WL1 × (0.5 × L1))
τM2= (WP × (L1 + L2)) + (WL1 × ((0.5 × L1) + L2)) + (WM1 × L2) + (WL2 × (0.5 × L2))
τM3= (WP × (L1 + L2 + L3)) + (WL1 × ((0.5 × L1) + L2 +L3)) + (WM1 × (L2 + L3))
+ (WL2 × ((0.5 × L2) + L3)) + (WM2 × L3) + (WL3 × (0.5 × L3))
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Motor Torque Calculation Example
Based upon arm configuration shown in the previous slide,
calculate the three motor torques, i.e. τM1, τM2, and τM3,
using the parameters in the table below:
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Titan gravity
1.373 m/sec2
Payload mass
2.0 kg
Linkage L1 length
0.3 m
Linkage L2 length
0.5 m
Linkage L3 length
0.6 m
Mass of Linkage L1
0.5 kg
Mass of Linkage L2
0.8 kg
Mass of Linkage L3
0.9 kg
Mass of Joint Motor 1
0.45 kg
Mass of Joint Motor 2
0.75 kg
τM1 = 0.927 Nm
τM2 = 3.226 Nm
τM3 = 7.340 Nm
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Gear Box
Selection:
a) A gear box will be
needed to achieved
desired output
motor torque
b) Gear Box Torque
Ratio is ratio of
output torque to
input torque, e.g.
100:1, 1000:1 etc.
c) E.g. for τM3 select
73.1 × 10-3 Nm ×
100:1 = 7.31 Nm
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Final Points
• Maximum holding torques calculated for
each motor. Would need to add contingency
(e.g. 20%) for each motor.
• Calibration, motion selection, and motor
torques also apply to a rover locomotion
chassis, i.e. steering and drive mechanism
• Same techniques can be applied, but chassis
details not covered here
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