Compared with a single discrete or serpentine arm, the discrete-serpentine heterogeneous dual-arm space robot (DSHDASR) combines the advantages of both kinds of arms, enabling it to perform complex on-orbit missions. The structural complexity of DSHDASR and cluttered environments pose a significant challenge in modeling and collision-free motion planning. To tackle the issue, this paper proposes a coordinated collision-free trajectory planning method for DSHDASR based on equivalent kinematics. Firstly, the mechanism and universal kinematics of DSHDASR are analyzed. Then, the equivalent kinematics model is established based on the spatial arc method. The whole kinematics chain of DSHDASR is described by the parameters of equivalent curves composed of space arcs. Furthermore, taking the target satellite transposition as an example, a coordinated collision-free trajectory planning is presented for DSHDASR based on the equivalent kinematics. The trajectory planning problem is formulated as the minimization of the objective function, which consists of kinematics constraint equations and obstacle avoidance constraint equations. The parameters of equivalent curves are obtained by optimizing the objective function, and the joint angles of DSHDASR can be determined using the above parameters. Finally, the mission of the target satellite transposition is simulated, and the results demonstrate the proposed method.