Inverse Kinematics 1 12212021 Inverse Kinematics given the

Inverse Kinematics 1 12/21/2021

Inverse Kinematics � given the pose of the end effector, find the joint variables that produce the end effector pose � for a 6 -joint robot, given find 2 12/21/2021

RPP + Spherical Wrist 3 12/21/2021

RPP + Spherical Wrist � solving 4 for the joint variables directly is hard 12/21/2021

Kinematic Decoupling � for 6 -joint robots where the last 3 joints intersecting at a point (e. g. , last 3 joints are spherical wrist) there is a simpler way to solve the inverse kinematics problem 1. use the intersection point (wrist center) to solve for the first 3 joint variables � 2. use the end-effector pose to solve for the last 3 joint variables � 5 inverse position kinematics inverse orientation kinematics 12/21/2021

RPP Cylindrical Manipulator d 3 d 2 d 1 oc zc yc xc 6 12/21/2021

RPP Cylindrical Manipulator d 3 d 2 d 1 oc zc yc xc 7 12/21/2021

RPP Cylindrical Manipulator d 3 d 2 d 1 oc zc yc xc 8 12/21/2021

RPP Cylindrical Manipulator d 3 d 2 d 1 oc zc yc xc 9 12/21/2021

RPP Cylindrical Manipulator d 3 d 2 d 1 oc zc yc xc 10 12/21/2021

RRP Spherical Manipulator 11 12/21/2021

RRP Spherical Manipulator 12 12/21/2021

RRP Spherical Manipulator 13 12/21/2021

RRP Spherical Manipulator 14 12/21/2021

RRP Spherical Manipulator 15 12/21/2021

Spherical Wrist Link ai ai di qi 4 0 -90 0 q 4 * 5 0 90 0 q 5 * 6 0 0 d 6 q 6 * * joint variable 16 12/21/2021

Spherical Wrist 17 12/21/2021

Spherical Wrist oc o d 6 18 12/21/2021

Inverse Kinematics Recap 1. Solve for the first 3 joint variables q 1, q 2, q 3 such that the wrist center oc has coordinates 2. Using the results from Step 1, compute Solve for the wrist joint variables q 4, q 5, q 6 corresponding to the rotation matrix 3. 19 12/21/2021

Spherical Wrist � for 20 the spherical wrist 12/21/2021

Spherical Wrist 21 12/21/2021

Spherical Wrist 22 12/21/2021

Spherical Wrist � if θ 5 23 =0 12/21/2021

Spherical Wrist � continued from previous slide only the sum θ 4+θ 6 can be determined 24 12/21/2021

Using Inverse Kinematics in Path Generation 25 12/21/2021

Path Generation �a path is defined as a sequence of configurations a robot makes to go from one place to another � a trajectory is a path where the velocity and acceleration along the path also matter 26 12/21/2021

Joint-Space Path �a joint-space path is computed considering the joint variables link 2 link 1 end effector path 27 12/21/2021

Joint-Space Path Joint Angles � linear joint-space path link 1 link 2 28 12/21/2021

Joint-Space Path � given the current end-effector pose and the desired final end-effector pose find a sequence of joint angles that generates the path between the two poses � idea � solve for the inverse kinematics for the current and final pose to get the joint angles for the current and final pose � interpolate the joint angles 29 12/21/2021

Joint-Space Path inverse kinematics 30 12/21/2021

Joint-Space Path find 0 Q from 0 T find f. Q from f. T t=1/m Q = f Q – 0 Q for j = 1 to m tj = j t j Q = 0 Q + t Q j set joints to j. Q end 31 12/21/2021

Joint-Space Path � linearly interpolating the joint variables produces �a linear joint-space path � a non-linear Cartesian path � depending on the kinematic structure the Cartesian path can be very complicated � some applications might benefit from a simple, or well defined, Cartesian path 32 12/21/2021

Cartesian-Space Path �a Cartesian-space path considers the position of endeffector link 2 end effector path link 1 33 12/21/2021

Cartesian-Space Path Joint Variable 1 � non-linear 34 joint-space path 12/21/2021

Cartesian-Space Path Joint Variable 2 � non-linear 35 joint-space path 12/21/2021

Issues with Cartesian-Space Paths 36 12/21/2021

Joint Velocity Issues � consider the RR robot shown below � assume that the second joint can rotate by ± 180 degrees 37 12/21/2021

Joint Velocity Issues � what happens when it is commanded to follow the straight line path shown in red? 38 12/21/2021

Joint Velocity Issues 39 12/21/2021

Joint Velocity Issues jump discontinuity in first derivative = infinite rotational acceleration steep slope = high rotational velocity 40 12/21/2021

Workspace � the reachable workspace of a robot is the volume swept by the end effector for all possible combinations of joint variables � i. e. , it is the set of all points that the end effector can be moved to 41 12/21/2021

Workspace � consider the RR robot shown below � assume both joints can rotate by 360 degrees 12/21/2021

Workspace � rotating the second joint through 360 degrees sweeps out the set of points on the dashed circle 12/21/2021

Workspace � rotating the first and second joints through 360 degrees sweeps out the set of all points inside the outer dashed circle 12/21/2021

Workspace � workspace consists of all of the points inside the gray circle 12/21/2021

Workspace � workspace consists of all of the points inside the gray circle 12/21/2021

Workspace � consider the RR robot shown below where the second link is shorter than the first � assume both joints can rotate by 360 degrees 12/21/2021

Workspace � rotating the second joint through 360 degrees sweeps out the set of points on the dashed circle 12/21/2021

Workspace � workspace consists of all of the points inside the gray area 12/21/2021

Workspace � consider the following straight line path shown in red � start point, end point, and all points in between are reachable 12/21/2021

Workspace � consider the following straight line path shown in red � start point and end point are reachable, but some points in between are not reachable 12/21/2021

Paths satisfying end point constraints 52 12/21/2021

Joint-Space Path �a joint-space path is computed considering the joint variables link 1 link 2 end effector path 53 12/21/2021

Joint-Space Path Joint Angles � linear joint-space path link 1 link 2 54 12/21/2021

Constraints � 55 12/21/2021

Velocity constraints � 56 12/21/2021

Acceleration constraints � 57 12/21/2021

Satisfying the constraints � 58 12/21/2021

Satisfying the constraints with polynomials � 59 12/21/2021

Satisfying the constraints with polynomials � 60 12/21/2021

Satisfying the constraints with polynomials � 61 12/21/2021

Example � 62 12/21/2021

Example: Joint angle cubic 63 12/21/2021

Example: Joint velocity quadratic 64 12/21/2021

Example: Joint acceleration linear 65 12/21/2021

Satisfying the constraints with polynomials � 66 12/21/2021

Satisfying the constraints with polynomials � 67 12/21/2021

Satisfying the constraints with polynomials � 68 12/21/2021

Example � 69 12/21/2021

Example: Joint angle quintic 70 12/21/2021

Example: Joint velocity quartic 71 12/21/2021

Example: Joint acceleration cubic 72 12/21/2021