# What is the difference between The state-space equation and The configuration-space equation

The craig(p.180) states that equations can be made to represent the dynamic equations. The first one being the state space equation.

$$\tau=M(\Theta)\cdot\ddot{\Theta} + V(\Theta,\dot{\Theta}) + G(\Theta)$$

The other one being the configuration state space equation.

$$\tau=M(\Theta)\cdot\ddot{\Theta} + B(\Theta)[\dot{\Theta} \dot{\Theta}]+ C(\Theta)[\dot{\Theta}^2] + G(\Theta)$$

Why would one need to split the velocity term into a Coriolis and centrifugal part? I need one of these versions to make a controller but can't see why one should be desirable than the other. Why is this important for computer control of a robot?

• What is the difference between $\dot{\Theta} \dot{\Theta}$ and $\dot{\Theta}^2$? – fibonatic Nov 29 '20 at 16:47
• That this is the angular velocities of the different joints. $\dot{\Theta}_{n}, \dot{\Theta}_{n+1}$ – Hamzalihi Nov 30 '20 at 16:38

From the Euler-Lagrange approach to the dynamics of a manipulator, you can build the $$B$$ and $$C$$ terms distinctively from the knowledge of your robot (e.g. distribution of masses of the links).
However, when it comes to designing the controller, you can conveniently group up terms in $$\Theta$$ and $$\dot{\Theta}$$ (first equation) as their influence can be aggregated.
Differently, if you don't have available the inertial parameters of your manipulator, you could think of estimating those dynamical terms, perhaps by means of adaptive control. In this specific case, you will most likely estimate $$V$$ rather than $$B$$ and $$C$$ separately, right because their effects accumulate and it is not straightforward to observe them in isolation.