## 1 Introduction  
The serial chain composed of rigid links connected by revolute joints is a reference model in robotics [1; 2] and musculoskeletal biomechanics [3; 4]. Closed-form dynamic equations (the dynamic equations that contain all the variables in explicit input-output form) have been derived for planar serial chains with a limited number (two or three) links, under additional simplifying assumptions related to both the passive and active elements of the chain [5; 6; 7; 8]. Often, the links are modeled as slender rods, and each joint is assumed to be driven by an abstract torque actuator that produces a torque about the joint axis with no effect on the joint reaction force. As the number of links increases, the complexity of the closed-form equations becomes prohibitive [4] (p. 394), and computations are usually performed iteratively, for one link at a time [1; 2; 4]. The resulting Newton-Euler equations are not in closed form, as the motion of a single link is kinematically coupled to the motion of the other links, and the contribution of individual variables can no longer be discriminated [1; 2; 4; 9].  
The joints of a musculoskeletal chain are driven by muscles that can span multiple joints and induce axial (traction and compressive) and shear joint reaction forces [10; 11; 12; 13]. Individual muscle actions, external forces acting on the chain, and inertial forces related to link movement can all contribute to joint torques and joint reaction forces [11; 14; 15]. Precise determination of the contribution that these unique components provide to the reaction forces acting on specific joint structures is one of the most crucial issues in musculoskeletal biomechanics [16; 17; 18].  
The aim of this study is to derive the _closed-form_ dynamic equations for a planar musculoskeletal chain, composed of a generic number \(n\) of rigid links of generic shape and size, and driven by single-joint and multi-joint muscle actuators that can span from one joint to all the joints of the chain. The specific goal is to determine the joint torque and the reaction force acting on each joint in _closed-form_ by analytical quantification of the unique contribution of each relevant biomechanical parameter: (1) the force of each single-joint or multi-joint muscle spanning or non-spanning the joint; (2) the weight and contact external resistances acting on each individual link of the chain; (3) the position, angular velocity, and angular acceleration of each individual link of the chain.  
The general closed-form equations derived in this study can be readily simplified to be applied to more basic chain models, for example, considering a limited number of links, replacing single and multi-joint joint muscle actuators with ideal torque actuators, and introducing simplifying assumptions on the link geometry and on the system of forces acting on each link. In particular, we will verify that the general closed-form equations can be reduced to the closed-form equations that have been reported for chains driven by ideal torque actuators and composed of two [4] or three [8] links modeled as slender rods, in absence of contact external resistances.  
The analytical results derived in this study can be applied to multilink musculoskeletal chains with deep/superficial and segmental/global muscles.