This report presents a theory of force control for computer-controlled manipulators, focusing on compliant motion. Compliant motion occurs when the manipulator's position is constrained by the task geometry. It can be achieved through passive mechanical compliance or active compliance implemented in the control loop. The report develops a formal model of force control based on the manipulator and task geometry, which provides a precise semantics for force control primitives in manipulator programming languages. These models are useful for two reasons: first, they define a simple interface between the manipulator and the programmer, isolating the programmer from the complexity of low-level control; second, they provide a method for synthesizing force control programs for compliant motion. The report discusses the development of a general method for control strategy synthesis in the ideal domain. The ideal domain is a model of the real world where the manipulator is represented by the ideal effector, and the task configuration is represented by equations relating the components of the ideal effector's force and velocity. The goal trajectory is the desired ideal effector position as a function of time, which must be consistent with the natural constraints. The control strategy synthesis problem is to find artificial constraints that will reproduce the desired goal trajectory given the natural constraints. The report also discusses the relationship between the ideal domain and the real world, and the transformation of task configurations and goal trajectories into natural constraints in the ideal domain. The synthesis of real-world control strategies involves three steps: modeling the task as natural constraints in the ideal domain, applying the synthesis method in the ideal domain to obtain artificial constraints, and transforming the artificial constraints into a corresponding real-world control strategy. The report concludes with a discussion of the relationship between this work and previous research in manipulation, including manipulator force control and automatic assembly planning. It also suggests further work in the field.This report presents a theory of force control for computer-controlled manipulators, focusing on compliant motion. Compliant motion occurs when the manipulator's position is constrained by the task geometry. It can be achieved through passive mechanical compliance or active compliance implemented in the control loop. The report develops a formal model of force control based on the manipulator and task geometry, which provides a precise semantics for force control primitives in manipulator programming languages. These models are useful for two reasons: first, they define a simple interface between the manipulator and the programmer, isolating the programmer from the complexity of low-level control; second, they provide a method for synthesizing force control programs for compliant motion. The report discusses the development of a general method for control strategy synthesis in the ideal domain. The ideal domain is a model of the real world where the manipulator is represented by the ideal effector, and the task configuration is represented by equations relating the components of the ideal effector's force and velocity. The goal trajectory is the desired ideal effector position as a function of time, which must be consistent with the natural constraints. The control strategy synthesis problem is to find artificial constraints that will reproduce the desired goal trajectory given the natural constraints. The report also discusses the relationship between the ideal domain and the real world, and the transformation of task configurations and goal trajectories into natural constraints in the ideal domain. The synthesis of real-world control strategies involves three steps: modeling the task as natural constraints in the ideal domain, applying the synthesis method in the ideal domain to obtain artificial constraints, and transforming the artificial constraints into a corresponding real-world control strategy. The report concludes with a discussion of the relationship between this work and previous research in manipulation, including manipulator force control and automatic assembly planning. It also suggests further work in the field.