Forward Kinematics

DH Representation

DH Representation - Frames (Convention) Frame i is attached to link i. The inertial frame is Frame 0 and Earth is link 0. Joint i joins links i-1 and i. (Another convention) The joint i+1 rotates about axis zi (DH1) The axis xi is perpendicular to the axis zi-1 (DH2) The axis xi intersects the axis zi-1 DH convention imposes two constrains thus enabling the use of only four parameters instead of six.

DH1

DH2

Assign frames at the two joints and at the end-point based on DH convention Write A1, A2, and A3 Frame i is attached to link i. The inertial frame is Frame 0 and Earth is link 0. Joint i joins links i-1 and i. The joint i+1 rotates about axis zi Link ai αi di 1 2

Three link cylindrical robot ai αi di i 1 2 3

Stanford Manipulator Link ai αi di i 1 2 3 4 5 6

SCARA Manipulator Link ai αi di i 1 2 3 4

Inverse Kinematics

Kinematic Decoupling Last three joints intersecting at a point (spherical wrist) Inverse position kinematics Inverse orientation kinematics Let

Find q1, q2, q3 such that is located at o. With q1, q2, q3 obtain Solve for q1, q2, q3 using the expression for from forward kinematic equations and its numerical value obtained from

Euler Angles

Velocity Kinematics Angular velocity of frame 2 wrt frame 0 velocity of p wrt frame 0

Angular Velocity

Linear Velocity

Velocity Kinematics - 2 DOF

Velocity Kinematics - 2 DOF

Singularities The values of q for which the rank of J decreases are called the singularities or singular configurations.

Singularities 2-DOF Velocity can be given in only x and y direction. when theta2 is zero. In that configuration arbitrary velocity in x and y direction cannot be imparted. One example is velocity in line with the manipulator when theta2 is zero.

Importance of Singularities certain direction of motion may be unattainable. bounded gripper velocities to unbounded joint velocities. bounded gripper forces and torques may correspond to unbounded joint torques. usually points on the boundary of the workspace. unreachable points in workspace under small perturbation of link parameters such as length, offset, etc. non-unique solution to inverse kinematics problem.

Inverse kinematics and Jacobian For desired end-point position and orientation as a function of time (the usual robot operation) X-dot can be evaluated and by inverting the Jacobian and so by integrating from a known initial value of q, other values of q can be obtained for given positions and orientations. This method is great but for singularities of the Jacobian.

Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.

Euler-Lagrange Equations Equations of motion for unconstrained system of particles is straightforward (F = m x a). For a constrained system, in addition to external forces, there exist constrained forces which need to be considered for writing dynamic equations of motion. To obtain dynamic equations of motion using Euler-Lagrange procedure we don’t have to find the constrained forces explicitly.

Holonomic Constraints

Nonholonomic Constraints General expression for nonholonomic constraints is: Nonholonomic constraints contain velocity terms which cannot be integrated out. A rear powered front steering vehicle

Consider a system of k particles, with corresponding coordinates, Often due to constraints or otherwise the position of k particles can be written in terms of n generalised coordinates (n < k), In this course we consider only holonomic constraints and for those constraints one can always find in principle n (n < k) independent generalised coordinates.

Virtual Displacements Define virtual displacements from above by setting dt=0. In our case dqj are independent and satisfy all the constraints. If they additional constraints have to be added to dqj to finally arrive at a statement of virtual displacements which have only independent dqj, we can replace dqj with

Virtual Work Let Fi be total force on every particle, then virtual work is defined as: Constraining forces do no work when a virtual displacement takes place (as is the case with holonomic constraints), so in equilibrium

D’Alembert’s Principle D’Alembert’s principle states that, if one introduces a fictitious additional force, the negative of the rate of change of particle i, then each particle will be in equilibrium.

Generalised Forces is called the i-th generalised force. The equations of motion become:

Euler-Lagrange Equations of Motion

Write the dynamic equations of motions for this system.

Expression for Kinetic Energy

Attach a coordinate frame to the body at its centre of mass, then velocity of a point r is given by:

Frame for I and Omega The expression for the kinetic energy is the same whether we write it in body reference frame or the inertial frame but it is much easier to write I in body reference frame since it doesn’t change as the body rotates but its value in the inertial frame is always changing. So we write the angular velocity and the inertia matrix in the body reference frame.

Jacobian and velocity D(q) is a symmetric positive definite matrix and is known as the inertia matrix.

Potential Energy V

Two Link Manipulator Links are symmetric, centre of mass at half the length.