Peter Ziegler.

This project builds a numerical inverse kinematics solver for a 6-DOF UR5 robotic arm. Given a desired position and orientation for the end-effector, the solver finds the six joint angles that put the arm there. I described the robot's geometry using a standard Denavit-Hartenberg parameter table, then wrote a forward kinematics routine that chains the six joint transforms together to compute where the end-effector currently is for any set of joint angles. The IK solver is iterative: at each step it computes the 6D error between the current and target poses, builds the Jacobian matrix that relates joint velocities to end-effector velocities, and uses that relationship to figure out a small joint update that should reduce the error. Repeat until the error drops below a tolerance. To validate it, I plugged the output joint angles into a CoppeliaSim UR5 model and confirmed the simulated arm lands at the desired pose.

The two parts I found most interesting to work through were the orientation error and the step size. Position error is easy because you can subtract two 3D vectors, but orientation error needs more thought since you can't subtract two rotation matrices in a meaningful way. I computed the rotation that takes the current orientation to the target, then pulled the rotation angle out using the trace of that matrix and the rotation axis out of its off-diagonal entries. That gave me a single 3D vector representing "how much to rotate, and around what axis" that fits into the same error format as position. The step size turned out to be more intresting than I expected. The full Newton step is fast near the solution but can overshoot when you're far from it, so scaling each update by a fraction makes the solver more stable at the cost of more iterations. I tested several values and plotted the convergence behavior. The full step converged in 9 iterations but bounced around at one point along the way, while smaller steps were slower but near completely linear.

This project performs an inverse dynamics analysis on a person's leg during gait, using marker motion capture data and ground reaction force measurements to calculate the internal forces and moments at the ankle, knee, and hip joints over time. I built a 2D model of the foot, leg, and thigh using standard anthropometric data to set each segment's length, mass, and moment of inertia from the subject's body weight. For each frame, the script tracks segment angles and center-of-mass positions from the marker data, numerically differentiates to get linear and angular velocities and accelerations, then works up the chain from foot to hip applying Newton-Euler equations at each joint. The output is a plot of joint moments versus time along with x and y intersegmental forces at the ankle, knee, and hip, which together describe the internal loading the body produces during the recorded motion.

The biggest challenge was that numerical differentiation amplifies noise dramatically, so the raw marker data couldn't be differentiated twice without producing wild accelerations. I handled this by applying a moving-average filter to the raw marker positions and then again after each derivative step, which kept the velocity and acceleration signals in a workable state. I also wrote two helper functions for central difference differentiation, one for paired x and y signals and one for single angular signals, which kept the main script readable and made the velocity and acceleration calculations consistent across all three segments. The other tricky part was getting the moment equations right at each joint, since the sign of the moment arm depends on which direction the segment's unit vector points and which end of the segment the force is applied at. To build confidence in the results, I chained the analysis from the ground up, using each joint's solved reaction force as the input to the next segment so that any error would show up clearly in the following plots.