By Zhijun Li

ISBN-10: 1447129628

ISBN-13: 9781447129622

ISBN-10: 1447129636

ISBN-13: 9781447129639

*Advanced keep watch over o**f Wheeled Inverted Pendulum Systems* is an orderly presentation of contemporary principles for overcoming the problems inherent within the keep watch over of wheeled inverted pendulum (WIP) platforms, within the presence of doubtful dynamics, nonholonomic kinematic constraints in addition to underactuated configurations. The textual content leads the reader in a theoretical exploration of difficulties in kinematics, dynamics modeling, complex keep watch over layout suggestions and trajectory new release for WIPs. a major trouble is tips to care for numerous uncertainties linked to the nominal version, WIPs being characterised by means of risky stability and unmodelled dynamics and being topic to time-varying exterior disturbances for which actual versions are challenging to return by.

The booklet is self-contained, providing the reader with every little thing from mathematical preliminaries and the fundamental Lagrange-Euler-based derivation of dynamics equations to numerous complicated movement regulate and strength keep watch over ways in addition to trajectory iteration technique. even though basically meant for researchers in robot regulate, *Advanced regulate of Wheeled Inverted Pendulum platforms *will even be helpful interpreting for graduate scholars learning nonlinear structures extra generally.

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**Extra resources for Advanced Control of Wheeled Inverted Pendulum Systems**

**Example text**

However, if f ∈ L1 ∩ L∞ , then f ∈ Lp for all p ∈ [1, ∞). Moreover, f ∈ Lp could not lead to f → 0 as t → ∞. If f is bounded can also lead to f → 0 as t → ∞. However, we have the following results. 24 (Barbalat’s Lemma) Consider the function φ : R+ → R. 25 Assume that a nonnegative scalar differentiable function f (t) enjoys the following conditions 1. d f (t) ≤ k1 f (t) dt ∞ 2. f (t) dt ≤ k2 0 for all t ≥ 0, where k1 and k2 are positive constants, then limt→∞ f (t) = 0. 81) f (s) ds ≥ f (0) − k1 k2 0 These inequalities imply that f (t) is a uniform bounded function.

62) The following theorem establishes the equivalence between the existence of a Lyapunov-like function and the input-to-state stability. 63) ∂V ∂V + f (t, x, u) ≤ −γ3 x ∂t ∂x where γ1 , γ2 and ρ are class K∞ functions and γ3 is a class-K function. 61) is ISS with γ = γ1−1 ◦ γ2 ◦ ρ. 65) for all t ≥ t0 . Define B = [t0 , T ) as the time interval before x(t) enters Rt0 for the first time. 68) for all t ∈ [t0 , ∞]. 63) is called an ISS Lyapunov function. 7 Lyapunov’s Direct Method This section presents an extension of the Lyapunov function concept, which is a useful tool to design an adaptive controller for nonlinear systems.

2) is a nonholonomic constraint that cannot be integrated analytically to result in a holonomic constraint among the configuration variables of the platform, namely xo , yo , and θ . As well known, the configuration space of the system is three dimensional (completely unrestricted), while the velocity space is two-dimensional. 4) where d is the platform width and r is the radius of wheel. Combining Eqs. 8) It is obvious that the matrix S(q) are in the null space of matrix A(q), that is, S T (q)AT (q) = 0.

### Advanced Control of Wheeled Inverted Pendulum Systems by Zhijun Li

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