Problems 1 Repeat Corke's Section 7.2.2.4 for the right arm of Baxter. Discuss any physical changes. Provide an introduction to Baxter's robot and identify its industrial applications 7.1 Forward Kinematics Forward kinematics is the mapping from joint coordinates, or robot configuration, to end-effector pose. We start in Sect. 7.1.1 with conceptually simple robot arms that move in 2-dimensions in order to illustrate the principles, and in Sect. 7.1.2 extend this to more useful robot arms that move in 3-dimensions. Fig. 7.1. a Mico 6-joint robot with 3-fin- gered hand (courtesy of Kinova Robotics). b Baxter 2-armed ro- botic coworker, each arm has 7 joints (courtesy of Rethink Robotics) a 7.2.2.4 Redundant Manipulator A redundant manipulator is a robot with more than six joints. As mentioned previ- ously, six joints is theoretically sufficient to achieve any desired pose in a Cartesian taskspace TCSE(3). However practical issues such as joint limits and singularities mean that not all poses within the robot's reachable space can be achieved. Adding additional joints is one way to overcome this problem but results in an infinite num- ber of joint-coordinate solutions. To find a single solution we need to introduce con- straints - a common one is the minimum-norm constraint which returns a solution where the joint-coordinate vector elements have the smallest magnitude. 211 7.3. Trajectories We will illustrate this with the Baxter robot shown in Fig. 7.1b. This is a two armed robot, and each arm has 7 joints. We load the Toolbox model >> mdl_baxter which defines two SerialLink objects in the workspace, one for each arm. We will work with the left arm >> left left = Baxter LEFT (Rethink Robotics) :: 7 axis, RRRRRRR, stdDH II theta d! alpha offset +- 1 11 1 21 1 31 1 1 51 1 61 171 911 921 931 941 951 961 971 0.271 01 0.364 01 0.3741 01 0.281 0.0691 01 0.069 01 0.01 01 01 -1.5711 1.571 -1.571 1.5711 -1.5711 1.571 01 01 1.5711 01 01 01 0 base: t = (0.064614,0.25858,0.119), RPY/xyz = [0, 0, 45) deg which we can see has a base offset that reflects where the arm is attached to Baxter's torso. We want the robot to move to this pose TE = SE3 (0.8, 0.2,-0.2) - SE.Ry (pi); which has its approach vector downward. The required joint angles are obtained us- ing the numerical inverse kinematic solution and >> q = left.ikine (TE) 0.0895 -0.0464 -0.4259 0.6980 -0.4248 1.0179 0.2998 is the joint-angle vector with the smallest norm that results in the desired end-effector pose. We can verify this by computing the forward kinematics or plotting >> left. fkine (q).print(\"xyz') t - (0.8, 0.2, -0.2), RPY/xyz - (180, 180, 180) deg 9 >> left.plot (9)