This paper presents a novel method for global motion planning of robotic systems that interact with the environment through contacts. The method directly handles the hybrid nature of such tasks using convex optimization. The motion-planning problem is formulated as a shortest-path problem in a graph of convex sets (GCS), where each node represents a convex set modeling quasi-static dynamics within a fixed contact mode. For each contact mode, semidefinite programming (SDP) is used to relax the nonconvex dynamics resulting from the simultaneous optimization of the object's pose, contact locations, and contact forces. This results in a tight convex relaxation of the overall planning problem, which can be efficiently solved and quickly rounded to find a feasible contact-rich trajectory. The method is evaluated on the task of planar pushing, where it consistently generates plans within a small percentage of the global optimum without requiring an initial guess. The method outperforms a state-of-the-art baseline for contact-rich planning, successfully finding trajectories where the baseline fails. The method is also demonstrated on a real robotic system. The approach is generalizable to more complex multi-contact problems. The method formulates the motion planning problem as a shortest-path problem in a GCS, where each node corresponds to a convex program for a specific contact mode. The method uses SDP relaxation to approximate the nonconvex dynamics and provides tight optimality bounds. The method is evaluated on two slider geometries, achieving high success rates and small optimality gaps. The method is compared to a contact-implicit trajectory optimization method, which relies on an initial guess and is less successful. The method is also tested on real hardware, demonstrating the feasibility of the generated motion plans. The method provides a tight convex relaxation of the planning problem, allowing for efficient solution and rounding to find feasible contact-rich trajectories. The method is able to generate plans that are close to globally optimal, with a mean upper bound on the optimality gap of around 10%. The method is able to handle complex multi-contact problems and is generalizable to a wide range of robotic tasks.This paper presents a novel method for global motion planning of robotic systems that interact with the environment through contacts. The method directly handles the hybrid nature of such tasks using convex optimization. The motion-planning problem is formulated as a shortest-path problem in a graph of convex sets (GCS), where each node represents a convex set modeling quasi-static dynamics within a fixed contact mode. For each contact mode, semidefinite programming (SDP) is used to relax the nonconvex dynamics resulting from the simultaneous optimization of the object's pose, contact locations, and contact forces. This results in a tight convex relaxation of the overall planning problem, which can be efficiently solved and quickly rounded to find a feasible contact-rich trajectory. The method is evaluated on the task of planar pushing, where it consistently generates plans within a small percentage of the global optimum without requiring an initial guess. The method outperforms a state-of-the-art baseline for contact-rich planning, successfully finding trajectories where the baseline fails. The method is also demonstrated on a real robotic system. The approach is generalizable to more complex multi-contact problems. The method formulates the motion planning problem as a shortest-path problem in a GCS, where each node corresponds to a convex program for a specific contact mode. The method uses SDP relaxation to approximate the nonconvex dynamics and provides tight optimality bounds. The method is evaluated on two slider geometries, achieving high success rates and small optimality gaps. The method is compared to a contact-implicit trajectory optimization method, which relies on an initial guess and is less successful. The method is also tested on real hardware, demonstrating the feasibility of the generated motion plans. The method provides a tight convex relaxation of the planning problem, allowing for efficient solution and rounding to find feasible contact-rich trajectories. The method is able to generate plans that are close to globally optimal, with a mean upper bound on the optimality gap of around 10%. The method is able to handle complex multi-contact problems and is generalizable to a wide range of robotic tasks.