Higher order loop constraints play a key role in the local mobility, singularity and dynamic analysis of closed loop linkages. Recently, closed forms of higher order kinematic constraints have been achieved with nested Lie product in screw coordinates, and are purely algebraic operations. However, the complexity of expressions makes the higher order analysis complicated and highly reliant on computer implementations. In this paper matrix expressions of first and second-order kinematic constraints, i.e. involving the Jacobian and Hessian matrix, are formulated explicitly for single-loop linkages in terms of screw coordinates. For overconstrained linkages, which possess self-stress, the first- and second-order constraints are reduced to a set of quadratic forms. The test for the order of mobility relies on solutions of higher order constraints. Second-order mobility analysis boils down to testing the property of coefficient matrix of the quadratic forms (i.e. the Hessian) rather than to solving them. Thus, the second-order analysis is simplified.

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