Understanding the Manipulator Equation

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My research goal is to build more capable robots that can apply explosive forces and accelerate like athletes. Basically, what I care about are forces and masses (inertia). Unfortunately, robots are quite complex. So, we need to up the level from the basics like F=ma, but the core concept does not differ too much from the Newton's law. 

Roboticists' version of newton's law is called the manipulator equation, which is probably the first scary equation you learn from robotics intro courses: 
M(q) \ddot{q} + c(q, \dot{q}) = J(q)^\top f + \tau.
I usually try to understand this equation by looking at the terms one by one and linking examples to them. 
1. M \ddot{q} is analogous to ma: larger mass makes it difficult to accelerate. 
2. c is a parasitic force required to maintain current pose q and velocity \dot{q}. For example, it is difficult to maintain squatted pose because you need to fight against the gravity. Another example is the Coriolis force you would feel when the car is making a rapid turn: you need some effort to keep yourself seated -  not only you need to press against the car door with your hand, but also your abs, hip, shoulder muscles twitch to remain seated. In fact, the c is often called the bias force - you need to offset system force by certain amount in contrast to no-offset-version of equation of motion (F=ma).   
3. J(q) has a scary name, Jacobian, but it is basically a matrix of lever arm lengths. So when you see J^\top f, just think about a seesaw: lever arm length is encoded within the J matrix and f is an external force (or weight) that presses the seesaw.  
4. \tau is actuators' torque. Think about an actuated revolving door and you are fighting against the door (J^\top f vs. \tau).

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