EVA Free-Flying Space Robot: Quintic Trajectories and Quaternion Attitude Control

The state is the base pose and velocity together with the arm joint vector $\mathbf{q}$. The floating-base equations of motion split into a base block and a manipulator block; the off-diagonal inertia block $\mathbf{H}_{bm}$ couples arm accelerations into the base. With $\mathbf{x}=[\mathbf{x}_b;\,\mathbf{q}]$ and $\mathbf{u}=[\mathbf{F}_b;\,\tau]$ (thruster wrench and joint torques):

With thrusters off and no external wrench, momentum is conserved, so every arm acceleration appears as base acceleration. The model therefore runs in free-flying mode: $\mathbf{F}_b$ from the thrusters is the control handle that cancels the reaction. The base attitude is the unit quaternion $\mathbf{q}=[\mathbf{q}_v;\,q_4]$ with kinematics and rotational dynamics

where $\tau_c$ is the thruster control torque and $\tau_d$ lumps the arm reaction and the grasp contact.

Each joint follows a rest-to-rest quintic polynomial with zero velocity and acceleration at both ends, so the reaction torque on the base also starts and ends at zero:

The extrema are

Scaling $T$ by a factor $k$ reduces the peak acceleration, and with it the peak base disturbance torque, by $k^2$. Approach and grasp are separate quintic segments, so the arm is at rest just before the gripper closes.

The regulator feeds back the vector part of the error quaternion and the body rate, which gives a near eigen-axis rotation without the Euler-angle singularity:

$\mathbf{K}$ and $\mathbf{C}$ are symmetric positive-definite. If the scalar part $q_{e4}$ is negative, the error quaternion is negated so the body takes the shorter rotation (no unwinding). For the Lyapunov candidate below, the cross terms cancel and LaSalle gives convergence to $\mathbf{q}_{ev}=0$ and $\omega=0$ for large-angle manoeuvres:

Gains are initialized from a natural frequency and damping ratio per axis, then refined against the thruster limits:

The regulator alone reacts after the base has moved, so a feedforward path is driven by the measured contact wrench. The wrench $\mathbf{F}_c$ is mapped to a base torque, low-pass filtered and scaled by a gain $\alpha\in[0,1]$, then added to the feedback torque:

Thrusters can only push, so opposing nozzles are used in pairs. The allocation block solves for non-negative forces with the smallest total impulse through the configuration matrix $\mathbf{B}$, and an on-off modulator (Schmitt trigger or pulse-width pulse-frequency stage) with a minimum on-time and a deadband turns each force demand into valve pulses. Propellant mass follows from the integrated impulse:

• Designed the dual-arm robot in SolidWorks (assembly Assem3) and imported it into Simscape Multibody with a 6-DOF joint to the world, so every arm reaction reaches the attitude loop.
• Generated the quintic joint references for the approach and grasp segments, with $q$, $\dot q$ and $\ddot q$ evaluated from the same duration $T$.
• Implemented the quaternion feedback regulator with the shortest-rotation logic.
• Built the force-regulator feedforward path from the gripper wrench to the thruster block, including allocation, modulation and impulse logging.

Three cases use the same arm trajectory and grasp timing. The entries are the expected behaviour from the physics; measured values from the Simscape runs are not included here.

• Logged metrics: peak attitude error and settling time in a 1 degree band, peak body rate, end-effector error at grasp, total thruster impulse ($I_{sp}$ about 60 to 70 s for cold-gas nitrogen, approximate), pulse count.
• Sanity checks: angular momentum constant with thrusters off, quaternion norm 1 within solver tolerance, quintic velocity and acceleration exactly zero at segment ends.

Rigid bodies, idealised sensing and actuation, a hand-chosen $T$, and a static target.

• Optimise $T$ for a limited reaction torque.
• Add sensor noise and thruster lag.