Fig. 3: Differentiable kinematic model.

a Two exemplified panels in kinematic analysis corresponding to the red dashed box in the subfigure below. The gray dot is fixed and the locations of the remaining colored nodes are solved, which can be divided into three basic geometry problems indicated by shades in different colors. b Details of three basic geometry problems to be solved sequentially. In each problem, nodes in colors can be analytically solved based on the fixed nodes (gray) and already-solved nodes (white). The solution of each problem can be considered as a transformation from known nodes to nodes to be solved. c Extracted forward computational graph (left) from Fig. 3b by composing the transformation in sequence. Different types of transformations are marked by arrows in different colors, pointing from preceding nodes to the updated nodes. Shaded regions represent geometry problems, from which the analytical expression of the transformations \({f}_{i}\) are obtained. A simplified graph is shown on the right. \({X}_{0}\) and \(\theta\) are the original node locations and deployed angle, respectively. \({X}_{i}\) is the set of updated nodes obtained from each transformation, corresponding to nodes of the same color in the original graph. d Backward computational graph to calculate the gradient of locations for the node of interest by composing the gradient of individual transformation \(\partial {f}_{i}\) via chain rules. e Sequential solving process to obtain all nodal locations. Only the right half is shown here as the left half is obtained by mirror symmetry on the center line. f Forward computational graph corresponding to sequential steps in Fig. 3e. The analytical transformations in each step are marked by arrows in different colors. Nodes that have been updated in previous steps are colored in white. g Backward computational graph (marked in red) to calculate the gradient of a given node.