One of the inherent problems of multi-limbed mobile robotic systems is the problem of multi-contact force distribution; the limbs in contact with its environment form closed kinematic chains, thus the contact forces and moments required to support it and those required by its tasks are indeterminate. A new computationally efficient analytical method for finding the optimal solution for the contact force distribution for a tethered multi-limbed robot climbing in unstructured environments is presented. The optimal solution is found by first finding the entire solution space for the contact force distribution for a statically stable stance under friction constraints, and then choosing an optimal solution in this solution space which maximizes the objectives given by the chosen optimization criteria. This approach will allow more options and freedom in choosing the final solution that not only satisfies the static equilibrium and friction constraints, but also can satisfy other special conditions under consideration at that instant. Finding the entire solution space first will give an intuitive visual map of how well the solution space is formed for the given conditions of the system, and choosing a solution in that space next will provide robustness against disturbances and indicate where this solution is positioned in the solution space to give insight into the quality of the chosen solution and the complex interactions of the many forces which are otherwise very difficult to see.

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