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ROBOTICS LAB · INTERACTIVE TUTORIAL 02 Reaching a point is only the beginning. A useful manipulator must follow a moving reference, correct its errors and avoid the objects around it. This follow-up turns our 3-DOF arm into a trajectory tracker—and asks what we can actually prove about its behaviour. In the first tutorial , we derived forward and inverse kinematics. Here we keep the same yaw–pitch–pitch geometry, derive its Jacobian, build a velocity controller and examine Lyapunov stability. Then we introduce spherical obstacles and a control-barrier safety filter. Trajectory tracking Whole-link obstacle checks Lyapunov & barrier functions 01 / TRY THE CONTROLLER Follow a path, then put something in the way Press Run simulation . The dashed curve is the desired tool path; the solid trace is what the robot achieves. The default sphere intersects the reference path, so perfect tracking and clearance cannot both be satisfied. Turn Obsta...

ROBOTICS LAB · INTERACTIVE TUTORIAL 01 A robot knows its joint angles. But where is its tool? Or, given a target in space, which angles should it choose? Explore both directions using a live three-degree-of-freedom manipulator. Yaw–pitch–pitch Position-only IK Four solution branches 01 / GEOMETRY AND TRANSFORMS Forward kinematics: from angles to position Consider a three-revolute-joint manipulator: a yawing base \(\theta_1\), a shoulder pitch \(\theta_2\), and an elbow pitch \(\theta_3\). The shoulder sits at height \(h\), and the two planar link lengths are \(L_1\) and \(L_2\). We ignore joint limits, tool offsets, and collisions for this analytical model. Using the standard Denavit–Hartenberg convention, a link transformation \(A_i\) is a product of a rotation about \(z\), a translation along \(z\), a translation along \(x\), and a rotation about \(x\). In homogeneous coordinates: Standard Denavit–Hartenberg transform \...