Exp[licit]-A Robot modeling Software based on Exponential Maps

TL;DR

This paper tackles the complexity of robot modeling by contrasting the traditional Denavit-Hartenberg (DH) convention with a differential-geometric approach based on exponential maps. It demonstrates how the Product of Exponentials and Adjoint-map formalisms enable a two-frame (S and ee) representation, reducing frame-management overhead and increasing modularity, while preserving accurate Forward Kinematics, Jacobians, and Mass Matrix construction. The authors present Exp[licit], a MATLAB toolbox that embodies the geometric method with a reusable RobotPrimitives core, symbolic support, visualization, and a Franka robot example, and show favorable computational performance against the DH-based Robotics, Vision and Control (RVC) toolbox. The work argues that differential-geometric methods offer practical benefits in flexibility, extensibility, and speed, enabling easier experimentation and potential real-time control in future C++/Python implementations. Overall, the paper provides both a conceptual framework and a practical software solution that can streamline open-chain robot modeling and control using exponential maps.

Abstract

$ $Deriving a robot's equation of motion typically requires placing multiple coordinate frames, commonly using the Denavit-Hartenberg convention to express the kinematic and dynamic relationships between segments. This paper presents an alternative using the differential geometric method of Exponential Maps, which reduces the number of coordinate frame choices to two. The traditional and differential geometric methods are compared, and the conceptual and practical differences are detailed. The open-source software, Exp[licit], based on the differential geometric method, is introduced. It is intended for use by researchers and engineers with basic knowledge of geometry and robotics. Code snippets and an example application are provided to demonstrate the benefits of the differential geometric method and assist users to get started with the software.

Figures (5)

• Figure 1: Computation times of RVC and Exp[licit] for five kinematic and dynamic calculations. For the Mass Matrix, Gravity, and centrifugal/Coriolis terms, the MEX-file option of RVC was invoked. First row: Comparison for Forward kinematics map (left) and Hybrid Jacobian (right) with respect to end-effector by using native Matlab scripts; Second row: Comparison for Mass Matrix of RVC (left) against Mass Matrix of RVC-MEX and Exp[licit] (right); Third row: Comparison for Gravity vector for RVC and RVC-MEX (left) against Gravity vector of Exp[licit] (right); Fourth row: Comparison for Coriolis Matrix of RVC (left) against Coriolis Matrix of RVC-MEX and Exp[licit] (right);
• Figure 2: Frames attached to an open-chain robot, using the DH-conventions.
• Figure 3: Franka robot at initial configuration. The DH-convention is shown in (A) and the geometric method in (B). Only two frames are required for the geometric method (B). The frames shown in (A) are derived from franka_url.
• Figure 4: Exp[licit] supports various 2D and 3D-robots. (A) Two planar robots: a Cart-Pole (left) and a Snake-Robot with variable DOF (right). (B) Two robots can be combined by using the addKinematics-method of the RobotPrimitives-class. In the example (B), the two robots of (A) are combined. (C) Currently supported 3D-robots: KUKA LBR iiwa (7 and 14 kg), YouBot, and Franka.
• Figure 5: Simulation of a simple impedance controller, using a Franka robot.