RMRRT: Riemannian Barrier Metric RRT for Inequality-Aware Steering on Equality Manifolds

This paper presents a motion planning framework that unifies equality and inequality constraints into a single geometric formulation for sampling-based planning in high-dimensional robotic systems. Conventional sampling-based planners often exhibit inefficient exploration because they treat constraints separately: equality constraints via projection and inequality constraints via binary collision checking. To address this issue, we propose Riemannian Barrier Metric RRT (RMRRT), which unifies both constraints into a single geometric formulation. RMRRT constructs a Riemannian barrier metric on the tangent space of the equality manifold by integrating the null-space projection of the constraint Jacobian with the Hessian of a logarithmic barrier function. This metric guides the tree expansion by selecting steering directions that minimize the induced Riemannian norm, thereby naturally avoiding inequality constraints while strictly adhering to the equality manifold. Experimental results demonstrate that RMRRT achieves a 100% success rate across diverse constrained manipulation tasks in both simulation and real-world experiments, while reducing planning time relative to state-of-the-art constrained planners. Ablation studies further show that the proposed barrier metric reduces sample rejection and shortens path length. Experiment videos and source code are available at: https://rcilab.github.io/rmrrt