Calculation Method of Static Friction Forces for Multi-Joint Manipulators

The static friction force is an important element of practical problems, such as the model-based control design of mechanical systems described by the Euler-Lagrange equations. The difficulty in calculating the static friction force, represented by a discontinuous function, lies in the fact that the differential equations representing the equations of motion become discontinuous differential-algebraic equations (DAEs). To solve the discontinuous DAEs using numerical methods, we need to solve a non-differential implicit algebraic equation at each step. In this letter, we propose an algorithm for calculating the static friction force by solving the implicit algebraic equation. Theoretical analysis shows that the friction force exists uniquely. This ensures that the proposed algorithm obtains a unique value for the static friction force. Moreover, the number of iterations in the proposed algorithm is only $3^{\mathrm { n}}$ at worst, where $\rm n$ denotes the number of manipulator joints. Hence, the proposed method matches numerical differential equation solvers, such as the Euler method. The effectiveness of the proposed method was validated through simulations using simple mechanical systems.