Introduction to ROBOTICS Manipulator Dynamics Manipulator Dynamics • Mathematical equations describing the dynamic behavior of the manipulator – For computer simulation – Design of suitable (model based) controllers – For evaluation and insight into the structure of the robot system – Joint forces/torques Robot motion, i.e. acceleration, velocity, position Forward/Inverse Dynamics Forward ; ; Dynamics Inverse Robot Motion Joint Forces/Torques Equations of Motion • The way in which the motion of the manipulator arises from forces/torques applied by the actuators, or from external forces/torques applied to the manipulator. Two Most Applied Approaches • Energy based: Lagrange-Euler Simple and symmetric • Momentum/force approach: Newton-Euler Efficient, derivation is simple but messy, involves vector cross product. Allows real time control Newton-Euler Algorithm • Newton-Euler method is described in next slides. The goal is to provide a big picture understanding of this method without getting lost in the details. • This approach includes two phases: Forward (Kinematic) Phase Backward (Kinetics) Phase Newton-Euler Algorithm • Forward Phase – First compute the angular velocity, angular acceleration, linear velocity, linear acceleration of each link in terms of its preceding link. – These values can be computed in recursive manner, starting from the first moving link and ending at the end-effector link. – The initial conditions for the base link will make the initial velocity and acceleration values to zero and g, respectively. Newton-Euler Algorithm • Backward Phase – Once the velocities and accelerations of the links are found, the joint forces can be computed one link at a time starting from the end-effector link and ending at the base link. Newton-Euler Algorithm; Forward Phase Angular Velocity Newton-Euler Algorithm; Forward Phase Angular Velocity Prismatic Joint: Revolute Joint: Newton-Euler Algorithm; Forward Phase Linear Velocity Newton-Euler Algorithm; Forward Phase Linear Velocity Prismatic Joint: Revolute Joint: Newton-Euler Algorithm; Forward Phase Angular Acceleration Newton-Euler Algorithm; Forward Phase Angular Acceleration Prismatic Joint: Revolute Joint: Newton-Euler Algorithm; Forward Phase Linear Acceleration Newton-Euler Algorithm; Forward Phase Linear Acceleration Prismatic Joint: Revolute Joint: Newton-Euler Algorithm; Forward Phase Linear Acceleration of center of mass For Both Revolute and Prismatic Joints: Newton-Euler Algorithm; Backward Phase Inertial Forces and Moments Inertia • If a force acts of a body, the body will accelerate. The ratio of the applied force to the resulting acceleration is the inertia (or mass) of the body. • If a torque acts on a body that can rotate freely about some axis, the body will undergo an angular acceleration. The ratio of the applied torque to the resulting angular acceleration is the rotational inertia of the body. It depends not only on the mass of the body, but also on how that mass is distributed with respect to the axis. Mass Distribution Inertia tensor- a generalization of the scalar moment of inertia of an object Moment of Inertia The moment of inertia of a solid body with density (r ) w.r.t. a given axis is defined by the volume integral I (r )r 2 dv, where r is the perpendicular distance from the axis of rotation. Moment of Inertia Moment of Inertia can be broken into components as: I jk mi ri 2 jk xi , j xi ,k for a discrete distribution of mass for a continuous distribution of mass i 2 I jk (r ) r jk x j xk dV V y2 z2 I ( x, y, z ) xy V xz xy z 2 x2 yz xz yz dxdydz. x 2 y 2 Moment of Inertia The inertia tensor relative to frame {A}: I xx I xy I xz A I I xy I yy I yz , I xz I yz I zz Mass moments of inertia x x z dv, y dv, I xx y 2 z 2 dv, V I yy I zz 2 V 2 V 2 2 Mass products of inertia I xy xydv, I xz xzdv, I yz yzdv. V V V Newton-Euler Algorithm; Backward Phase Force Equation Newton-Euler Algorithm; Backward Phase Moment Equation Newton-Euler Algorithm; Backward Phase Joint Force/Torque Prismatic Joint: Newton-Euler Algorithm; Backward Phase Joint Force/Torque Revolute Joint: Inclusion of Gravity Forces • The effect of gravity loading on the links can be included by setting , where g is the gravity vector. The Structure of the Manipulator Dynamic Equations M (q)q C(q, q) G(q) M (q) : n n C (q, q ) : n 1 G (q ) : n 1 : mass matrix : centrifugal and Coriolis terms : gravity terms The Structure of the Manipulator Dynamic Equations The Structure of the Manipulator Dynamic Equations The Structure of the Manipulator Dynamic Equations • G(q)=?