MAE204 β€” Mobile Manipulation with the youBot

Feedforward + PI task-space control for pick-and-place on a mobile manipulator

πŸ“„ Full Report (PDF)

This page covers the lab assignments and final project from MAE 204: Introduction to Robotics at UC San Diego, all using the UR3e robot arm and the Modern Robotics framework.

Lab 1 β€” Introduction to the UR3e

First contact with the physical UR3e robot (6-joint tabletop arm, 11 kg, 3 kg payload, Β±360Β° joint limits). The robot was controlled entirely via the teach pendant β€” no code, no scripting.

The task was the classic Tower of Hanoi: three stacked blocks on tower A must be moved to another tower, one block at a time, without placing a larger-numbered block on a smaller one. Each pick-and-place move was programmed by physically jogging the arm to each position and recording it as a waypoint (Basic > Waypoint > Set Waypoint) on the pendant. The standby configuration was fixed at joint angles [-30, -90, 90, -90, -90, 150] degrees.

Lab 2 β€” Forward Kinematics

Implemented forward kinematics for the physical UR3e robot arm using the product of matrix exponentials:

\[T(\theta) = e^{[S_1]\theta_1} e^{[S_2]\theta_2} \cdots e^{[S_6]\theta_6} M\]

The six screw axes $S_1 \ldots S_6$ and the zero-configuration end-effector matrix $M$ were derived by hand from physical measurements of the robot. The MATLAB script converts joint angles (degrees) to the end-effector SE(3) transformation and prints position in mm. Results were verified in the lab by commanding the real UR3e to specific joint configurations and measuring the end-effector position.

Lab 3 β€” Inverse Kinematics

Implemented numerical inverse kinematics via the Newton-Raphson method with the Moore-Penrose pseudoinverse of the body Jacobian (IKinSpace from the Modern Robotics library), extended with joint-limit wraparound correction to keep all angles within Β±360Β°.

The lab scenario: an Evil Dragon is invading UCSD and threatening the coffee supply. The only defense is to assemble the Effigy of Graduate Student-Warrior β€” three scattered blocks (legs, body, head) β€” using the UR3e with a Robotiq Hand-E gripper. The IK script computed a 15-waypoint sequence of SE(3) gripper poses, solved for joint angles at each waypoint, and output a waypoint_array.csv that drove the real robot to pick and stack the blocks in order.

Final Project β€” Mobile Manipulation with the youBot

This project implements a complete software stack to control a youBot mobile manipulator performing a pick-and-place task in CoppeliaSim Scene 6: driving to a cube, grasping it, and placing it at a target location.

System Architecture

The pipeline is organized into four components:

  1. NextState β€” A first-order Euler simulator propagating the 12-variable robot configuration (chassis pose + 5 arm joints + 4 wheel angles) given commanded velocities. Chassis odometry uses the exact body-twist integration from Modern Robotics Ch. 13.4; joint/wheel speeds are clipped to a configurable maximum.

  2. TrajectoryGenerator β€” Produces an 8-segment reference trajectory for the end-effector using ScrewTrajectory from the Modern Robotics library:
    • Move to standoff above cube β†’ Descend to grasp β†’ Close gripper β†’ Lift to standoff
    • Move to goal standoff β†’ Descend to place β†’ Open gripper β†’ Return to standoff
  3. FeedbackControl β€” Implements the feedforward + PI control law (MR Eq. 11.16 / 13.37):
\[\mathcal{V}(t) = [\text{Ad}_{X^{-1}X_d}]\,\mathcal{V}_d + K_p\, X_{\text{err}} + K_i \int_0^t X_{\text{err}}\,dt\]

where $X_{\text{err}} = \log(X^{-1}X_d)$. The commanded twist $\mathcal{V}$ is mapped to wheel and arm joint speeds via the pseudoinverse of the $6\times9$ mobile-manipulator Jacobian $J_e(\theta)$.

  1. Run script β€” Ties all components together, outputs a CSV file playable in CoppeliaSim.

Engineering Decisions

Singularity avoidance. The end-effector approaches at a 45Β° angle from vertical rather than horizontally. A horizontal approach placed the arm near full extension at the grasp pose, producing a near-singular Jacobian and unreasonably large joint speeds. The 45Β° tilt keeps the arm well-conditioned. Pseudoinverse tolerance is set to rcond=1e-3 to suppress near-zero singular values.

Self-collision avoidance. Certain cube goal orientations caused the arm to fold backward over the chassis. Inspecting offending CSV files revealed joints well outside physical limits (e.g., ΞΈβ‚„ = βˆ’3.8 rad vs. limit Β±1.79 rad). The Jacobian-column-zeroing method from MR Capstone was evaluated but degraded tracking enough to cause grasp failures. The adopted solution: careful task-geometry selection combined with the pseudoinverse tolerance to keep the arm in its forward workspace.

Results

Best Case β€” Feedforward + P Control

Gains: $K_p = 3I$, $K_i = 0$
Initial config: $(\phi, x, y) = (\pi/6, -0.3, 0.1)$, 40.1Β° orientation error, 0.337 m position error

All six error components converge smoothly and monotonically to zero within ~2 seconds, with no overshoot. The error constant of ~0.33 s ensures full convergence well before the grasp at t β‰ˆ 5 s.

Best case: end-effector error X_err (left) and manipulability μ₁(AΟ‰), μ₁(Av) (right) vs. simulation time. Smooth convergence with no overshoot.

Overshoot Case β€” Feedforward + PI Control

Gains: $K_p = 2I$, $K_i = 8I$

The aggressive integral gain causes clear overshoot and oscillation in Ο‰β‚“, ωᡧ, and vᡧ components. The integrator accumulates past error and overshoots zero, producing an underdamped transient that settles by t β‰ˆ 4 s β€” still in time for a successful grasp.

Overshoot case: oscillatory transient from the aggressive integral term (K_i = 8I). Error settles before the grasp phase.

New Task β€” Custom Workspace Configuration

Gains: $K_p = 5I$, $K_i = 1I$
Initial config: $(\phi, x, y) = (0.3, -0.3, -0.3)$, 31.9Β° orientation error, 0.240 m position error
Cube: initial $(0.5, -0.6, 0)$ β†’ goal $(1.0, 0.5, -\pi/4)$
Video: (coming soon)

The cube is picked up from the right side of the workspace and placed at a βˆ’45Β° rotation in the forward-left area. Configuration is loaded from an external YAML file (newtask_config.yaml). All six error components decay monotonically β€” smooth convergence with no overshoot.

New task: pick-and-place with a custom cube placement geometry. Monotonic convergence with well-tuned PI gains.

Key Takeaways

  • Integral term trade-off: Eliminates steady-state error from modeling biases but causes overshoot as the accumulated integral drives the system past the reference. Higher K_i worsens oscillation.
  • Manipulability: μ₁(Av) drops near 0.1 during grasp/place approach (arm near full extension β†’ near-singular Jacobian), then recovers during transport. μ₁(AΟ‰) stays near 0.31 during steady-state motion.
  • Velocity vs. torque control: Velocity control is adequate for free-space pick-and-place but insufficient for contact-rich tasks (surface polishing, peg-in-hole) where force/torque feedback and the full dynamic model (mass matrix, Coriolis terms) are required.