###### Abstract  
This paper studies a cooperative modeling framework to reduce the complexity in deriving the governing dynamical equations of complex systems composed of multiple bodies such as biped robots and unmanned aerial and ground vehicles. The approach also allows for an optimization-based trajectory generation for the complex system. This work also studies a fast-slow model predictive control strategy with task prioritization to perform docking maneuvers on cooperative systems. The method allows agents and a single agent to perform a docking maneuver. In addition, agents give different priorities to a specific subset of shared states. In this way, overall degrees of freedom to achieve the docking task are distributed among various subsets of the task space. The fast-slow model predictive control strategy uses non-linear and linear model predictive control formulations such that docking is handled as a non-linear problem until agents are close enough, where direct transcription is calculated using the Euler discretization method. During this phase, the trajectory generated is tracked with a linear model predictive controller and addresses the close proximity motion to complete docking. The trajectory generation and modeling is demonstrated on a biped robot, and the proposed MPC framework is illustrated in a case study, where a quadcopter docks on a non-holonomic rover using a leader-follower topology. c 2024243189 202424318  
The complexity of the problem is reduced in multiple ways. One way to carry this out is to use low-fidelity dynamic models for the robot such as single rigid body dynamics model, where lumped inertia is attached to the body frame and actuated links are assumed to be moving slowly and having low inertia [2]. Another way to reduce complexity is to split the the optimization into smaller sub-problems and to pre-define some portions of the trajectory such as footholds [9]. However, simplifications in robot model and optimization problems are compromised with the complexity of the motion that can be calculated with trajectory optimization. For this reason, having means to simplify full-body dynamics without compromising the fidelity of the model is vital. Recently, the trajectory optimization problem was distributed into smaller alternating sub-problems, where one first satisfies dynamic constraints of the problem by finding robot momentum and contact forces, and then one finds the leg kinematics that satisfy robot dynamics. This is denoted as centroidal and whole-body model splitting and was first introduced in the work [8], where a sequential optimal control formulation was represented. In another work based on the same splitting of whole-body motion and centroidal dynamics, the locomotion problem is cast into a mathematical framework based on Alternating Direction Method of Multipliers (ADMM) [10]. This paper exploits the natural splitting between centroidal and manipulator dynamics and ensures consensus between these models. Another recent example uses the same splitting and introduces an accelerated ADMM algorithm to solve the locomotion problem. [3]. Besides the centroidal and whole body splitting of the dynamic model of a legged robot, there is obviously another split between the portions of the body. This splitting is naturally defined by the design of legged robots and referred to as body and limbs. Depending on the design, they can contain a single body or a set of multi-bodies and interconnections between these sets maintained over a connection node, which is a joint. A formal way to describe the cooperation between these sets already exists in the literature within a cooperative system framework [11]. This framework defines a relationship between independent agents using communication graphs, which dictate a mapping to the information flow among agents.