7 Class Problem: Spherical Wrist 1. Fill in the table of D-H parameters for the spherical wrist.2. write the three D-H transformation matrices (one for each joint) for the spherical wrist3. Find the overall transformation matrix which relates the final coordinates (x6y6z6) to the “base” coordinates (x3y3z3) for the spherical wrist
9 Joint Space and Operational Space Description of end-effector taskposition: coordinates (easy)orientation: (n s a) (difficult)w.r.t base frameFunction of timeOperational spaceJoint spacePrismatic: dRevolute: thetaIndependent variables
10 Joint Space and Operational Space Direct kinematics equationThree-link planar arm(Pp )M的最大值是？平面运动只需要3个自由度
11 Joint Space and Operational Space Generally not easy to express
12 Joint Space and Operational Space Workspacereachable workspacedexterous workspaceFactors determining workspaceManipulator geometryMechanical joint limitsMathematical description of workspaceWorkspace is finite, closed, connected
14 Performance Indexes of Manipulator Accuracy of manipulatorDeviation between the reached position and the position computed via direct kinematics.repeatability of manipulatorA measure of the ability to return to a previously reached position.
15 Kinematic Redundancy Definition A manipulator is termed kinematically redundant when it has a number of degrees of mobility which is greater than the number of variables that are necessary to describe a given task.
16 Kinematic Redundancy Intrinsic redundancy m<n functional redundancy relative to the taskWhy to intentionally utilize redundancy?
17 Kinematic Calibration Kinematic calibration techniques are devoted to finding accurate estimates of D-H parameters from a series of measurements on the manipulator’s end-effector location.Direct measurement of D-H is not allowed.
19 Inverse Kinematicswe know the desired “world” or “base” coordinates for the end-effector or toolwe need to compute the set of joint coordinates that will give us this desired position (and orientation in the 6-link case).the inverse kinematics problem is much more difficult than the forward problem!
20 Inverse Kinematicsthere is no general purpose technique that will guarantee a closed-form solution to the inverse problem!Multiple solutions may existInfinite solutions may exist, e.g., in the case of redundancyThere might be no admissible solutions (condition: x in (dexterous) workspace)
21 Inverse Kinematicsmost solution techniques (particularly the one shown below) rely a great deal on geometric or algebraic insight and a few common “tricks” to generate a closed-form solutionNumerical solution techniques may be applied to all problems, but in general do not allow computation of all admissible solutions