Robotics: Dynamics, Control and Motion planning (Part 2)

Understand the link between end effector velocities and joint velocities via forward and inverse differential kinematics using the Jacobian, and address singularities with damped least squares and null space control.

Compute end effector velocities from joint velocities using the Jacobian in differential kinematics for two and three DoF planar manipulators, covering linear and angular motion.

Explain the velocity propagation model for serial manipulators, showing how angular velocity propagates through revolute joints and how linear velocity accumulates, yielding end-effector motion from the Jacobians Jv and Jomega.

Compute end effector linear velocity and angular velocity for a two-link planar serial manipulator from joint velocities using velocity propagation and Jacobian representation in the x–y plane.

It demonstrates velocity propagation for a four-joint Scara manipulator, including a prismatic joint, using forward kinematics and the six-by-four Jacobian to map joint velocities to end-effector velocities.

Singularities in manipulators arise when jacobian determinant is zero, causing loss of degrees of freedom and velocities due to gimbal lock; mitigate with redundancy, path planning, and damped least squares.

Explore how Euler-Lagrange dynamics model robotic manipulators, deriving the inertia matrix, Coriolis terms, and gravity vector to compute joint torques via M(q) qddot + C(q,qdot) qdot + G(q) = tau.

Compute the dynamic equation for a three degree of freedom planar manipulator with three revolute joints, deriving inertia, centrifugal, gravity terms, and joint torques via the Lagrange method.

Apply the Newton-Euler formulation to robot dynamics, using forward recursion to compute link velocities and accelerations and backward recursion to derive forces and torques at each link.

Calculate forward recursion for a two-link RR serial manipulator, deriving angular velocity, angular acceleration, and linear acceleration of joints and centers of mass under gravity.

Derive backward recursion for a two-link rr manipulator, compute forces and moments on each link, and obtain joint torques via Newton Euler formulation, highlighting its efficiency over Lagrange Euler formulation.

apply lagrangian dynamic formulation to an rp manipulator with one revolute and one prismatic joint, derive equations of motion with euler-lagrange and compute kinetic and potential energies.

Compute the Plücker coordinates and moment of a line from p and d, then derive the twist and wrench along the screw axis.

Explore trajectory planning for robotic manipulators, detailing path planning, time parameterization, and joint space versus task space approaches to generate smooth, time-resolved end effector trajectories.

The work cell controller coordinates conveyors, sensors, robots and accessories to execute tasks safely and efficiently. It handles safety, data communication, error handling, and optimization, including energy saving measures.

Explore open loop and closed loop manipulator control and diverse methods—PD, PID, torque, adaptive, impedance, hybrid position/force, motion-rate, and LQR—with industrial, surgical, and collaborative robot applications.

Apply pid control to drive a joint from 20° to 30° using kp=10, ki=5, kd=2, with integral error zero and derivative -5 deg/s, yielding a torque of 90 newton meters.

Learn to compute an LQR gain for a simple linear system by solving the algebraic Riccati equation, deriving the gain k, and applying u = -k x.