Geometric Algebra for Optimal Control with Applications in Manipulation Tasks

TL;DR

The paper proposes geometric algebra (GA), and in particular conformal GA, as a unifying framework for robot geometry that seamlessly integrates kinematics, dynamics, and optimization. It extends serial-manipulator dynamics to include a position-dependent inertia tensor, and formulates cost functions for IK and optimal control directly in the motor and primitive representations, enabling uniform handling of points, lines, planes, and higher-order primitives. A hands-on contribution is gafro, an open-source library with a fast CGA-based implementation that achieves faster kinematics than traditional libraries and supports MPC/iLQR-style optimization with GA-based costs. The experiments on a Franka Emika robot demonstrate accurate inverse dynamics control, fast IK solutions, and versatile reaching tasks across multiple geometric primitives, highlighting GA's practical impact for real-time robotic manipulation and planning.

Abstract

Many problems in robotics are fundamentally problems of geometry, which lead to an increased research effort in geometric methods for robotics in recent years. The results were algorithms using the various frameworks of screw theory, Lie algebra and dual quaternions. A unification and generalization of these popular formalisms can be found in geometric algebra. The aim of this paper is to showcase the capabilities of geometric algebra when applied to robot manipulation tasks. In particular the modelling of cost functions for optimal control can be done uniformly across different geometric primitives leading to a low symbolic complexity of the resulting expressions and a geometric intuitiveness. We demonstrate the usefulness, simplicity and computational efficiency of geometric algebra in several experiments using a Franka Emika robot. The presented algorithms were implemented in c++20 and resulted in the publicly available library \textit{gafro}. The benchmark shows faster computation of the kinematics than state-of-the-art robotics libraries.

Figures (13)

• Figure 1: A generic cost function in geometric algebra can consider different geometric primitives without changing its structure.
• Figure 2: Non-zero elements of various geometric primitives in their primal representations in conformal geometric algebra. Boxes represent basis blades and colored boxes represent the non-zero blades of the geometric primitive with the matching color. It can be seen that of the 32 basis blades only a sparse number is used for the representations. Note that geometric primitives are single-grade objects, while transformations are mixed-grade. The boxes correspond to the basis blades that are shown in Table \ref{['tab:basis_blades_of_conformal_geometric_algebra']} in Appendix \ref{['sub:structure_of_a_multivector_in_conformal_geometric_algebra']}.
• Figure 3: Rigid body transformations of various geometric primitives. Red marks the initial primitive and green the final one, the primitives resulting from the trajectory of interpolated motors are shown in blue.
• Figure 4: Benchmarking results for gafro compared to Raisim, Pinocchio and KDL. The benchmarks were all performed on an AMD Ryzen 7 4800U CPU using the compiler flags -O3 -msse3 -march=native. The presented results are the average of 10000 executions with 10 repetitions.
• Figure 5: Optimal trajectory for an oriented pointmass system. We placed four target motors along the trajectory at $T/4$, $T/2$, $3T/4$ and $T$, respectively. The target motors are highlighted along the trajectory. The optimal trajectory was then found using the system defined in Equation \ref{['eq:bivector_linear_system']} and the cost function from Equation \ref{['eq:motor_ik_manifold']}.