By Vincent Duindam

Walking robots are complicated machines with many levels of freedom. Designing effective controllers for such robots could be a daunting job, and the differential equations by way of themselves often don't support a lot whilst attempting to comprehend the dynamics. nonetheless, examine on passive dynamic strolling robots has proven that it's attainable to make robot mechanisms stroll very evidently and successfully with out utilizing any keep watch over! the distance among theoretically well-understood position-controlled jogging robots and experimentally-designed out of control passive-dynamic walkers is however huge, and increasing a passive-dynamic walker to be extra powerful and flexible is non-trivial.

The function of this paintings is to offer a collection of mathematical instruments which may simplify learning robot jogging motions and designing energy-efficient controllers. The authors expand classical dynamic modeling tools and look at robots and controllers as energy-exchanging actual platforms, which varieties the basisof the so-called port-based procedure. They express how such equipment can be utilized to investigate jogging mechanisms, locate effective strolling trajectories, and layout controllers that raise robustness and balance with minimum strength price. large examples and illustrations are used with the target to make the math intuitive and obtainable to every person with an engineering background.

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**Additional resources for Modeling and Control for Efficient Bipedal Walking Robots: A Port-Based Approach**

**Example text**

The relation between the forces acting on a mechanism and its motion. 1 Forces on Rigid Mechanisms We ﬁrst look at the eﬀect of forces on rigid bodies. We want to describe these forces in such a way that the power P associated with the action of forces on a rigid body with a twist T can be formulated as P = W T T , just like we have P = f T v for point masses with velocity v and force f . Thus, we look for an expression W describing somehow the forces as a vector which, when paired with the twist, gives the power.

K + 1)! 9. From the expression, we can see that S(Q, 0) = I, and we can evaluate the partial derivative with respect to φ at φ = 0 as follows. ∂(S T (Q, φ)p) ∂φ = φ=0 ∂ ∂φ p− 1 T 1 adφ p + (ad2φ )T p − . . 2 6 φ=0 ∂ = ∂φ I0 0I 1 φ˜a φ˜b pa + pb 2 0 φ˜a pa + ... pb ∂ = ∂φ I0 0I 1 p˜a p˜b pa − pb 2 p˜b 0 φa + ... 83) which holds for all vectors x, y ∈ R3 . Since we used the twist in body coordinates and ignored gravity, the inertia tensor I 0,0 and the Hamiltonian do not depend on Q, and hence not on φ either.

0: choose α = ω , ω ˆ= ω , r = α1 ω ˆ ∧ (v − ω ˆ T vω ˆ ), and β = ω ˆ T v. With these choices of α, β, ω ˆ , and r, any twist can be expressed in the form of the theorem. 8, we can make a coordinateindependent distinction between a purely translational motion (α = 0), a rotational motion (β = 0), or a general so-called screw-motion (α = 0, β = 0). The ratio of the numbers α and β is called the pitch. Finally, let us present some useful identities about twists and transformation matrices that are used later in this book.