Controlled Motion of a Spherical Robot with Feedback. II

Abstract

In this paper, we develop a model of a controlled spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum, with a feedback that stabilizes given partial solutions for a free system at the final stage of motion. According to the proposed approach, feedback depends on phase variables (current position, velocities) and does not depend on the specific type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulations of motion with feedback that demonstrate the efficiency of the proposed model.

Appendix

We determine the number of real roots of the polynomial (13)

$$P_{4}(\lambda) = a\lambda^{4} + bk_{\vartheta}\lambda^{3} + c\lambda^{2} + dk_{\vartheta}\lambda + e $$

depending on the values of $k_{\vartheta }$ and $\omega _{0}$. To do so, we use the Sturm method [19] for finding the number of real roots among all roots of the equation.

Let us call a sequence of signs corresponding to values of the Sturm polynomials, $f_{0},\ldots , f_{4}$, with a chosen value of argument λ a series of signs:

$$[{\text{sign}}f_{0}(\lambda),{\text{sign}}f_{1}(\lambda),{\text{sign}}f_{2}(\lambda),{\text{sign}}f_{3}(\lambda),{\text{sign}}f_{4}(\lambda)]. $$

According to the Sturm theorem, the number of real roots N is determined as the difference

$$ N=W(\lambda_{-})-W(\lambda_{+}), $$

(24)

where $\lambda _{-}$ and $\lambda _{+}$ are, respectively, a negative and a positive value of $\lambda$ sufficiently large in absolute value, and $W(\lambda )$ is the number of sign changes in the corresponding series of signs.

It is obvious that

$$\begin{array}{cc} &{\text{sign}}f_{i}(\lambda_{-})=(-1)^{i}\,{\text{sign}}\,A_{i}(k_{\vartheta},\omega_{0}),\\ &{\text{sign}}f_{i}(\lambda_{+})={\text{sign}}\,A_{i}(k_{\vartheta},\omega_{0}), \quad i = 0,\ldots,4, \end{array} $$

where $A_{i}(k_{\vartheta },\omega _{0})$ are the coefficients at higher degrees of the Sturm polynomials $f_{i}(\lambda )$. We write them in explicit form (up to multiplication by positive constants):

$$ \begin{array}{c} A_{0}(k_{\vartheta},\omega_{0})= 1, \quad A_{1}(k_{\vartheta},\omega_{0})= 1, \quad A_{2}(k_{\vartheta},\omega_{0})= 3b^{2}k_{\vartheta}^{2}-8ac,\\ A_{3}(k_{\vartheta},\omega_{0})=-3b^{3}d k_{\vartheta}^{4} - k_{\vartheta}^{2}(18a^{2} d^{2}+ 6ab^{2}e-14abcd-b^{2}c^{2})+ 4ac(4ae-c^{2}),\\ A_{4}(k_{\vartheta},\omega_{0})=-4b^{3}d^{3}k_{\vartheta}^{6}-k_{\vartheta}^{4}(27a^{2}d^{4}+ 6ab^{2}d^{2}e -18abcd^{3}+ 27b^{4}e^{2}-\\ -18b^{3}cde-b^{2}c^{2}d^{2}) -4k_{\vartheta}^{2}(48a^{2}bde^{2}-36a^{2}cd^{2}e-36ab^{2}ce^{2}+ 20abc^{2}de+\\ +ac^{3}d^{2}+b^{2}c^{3}e)+ 16ae(4ae-c^{2})^{2}, \end{array} $$

(25)

where $a, b, c(\omega _{0}), d, e$ are the constants defined above in Eqs. (8) and (14).

We define regions where the coefficients $A_{i}(k_{\vartheta },\omega _{0}),\, i = 2,\ldots ,4$ have a constant sign (A ₀ > 0 and $A_{1}>0$ do not depend on the values of $k_{\vartheta }, \omega _{0}$). These regions are shown as a diagram in Fig. 4 constructed for parameter (22). The boundaries of the regions are defined as roots of the equations $A_{i}(k_{\vartheta },\omega _{0})= 0,\, i = 2,\ldots ,4$.

Fig. 4

Regions where the functions A _i (ω ₀,k _𝜗 ) (25) on the parameter plane (ω ₀,k 𝜗2) have a constant sign. The right panel shows an enlarged fragment of the left picture

For each of these regions we write series of signs with $\lambda =\lambda _{\pm }$, determine the number of sign changes $W(\lambda _{-}), W(\lambda _{-})$ and the corresponding number of real roots N (24) (see Table 1).

Table 1 Series of signs and the number of real roots N of Eq. (13)

Thus, three different cases are possible.

• 1. For parameters (ω ₀,k _𝜗 ) from regions I, II, and IV in Fig. 4, Eq. (13) has no real roots. It has as its roots two pairs of conjugate complex numbers with a negative real part. Consequently, for parameters $(\omega _{0}, k_{\vartheta })$ from regions I, II, and IV the family of fixed points (2) of the reduced system (1) with additional control of the form (11) is of stable focus type.

• 2. For parameters (ω ₀,k _𝜗 ) from regions III and V in Fig. 4, the solution of Eq. (13) will be a pair of real negative roots and a pair of conjugate complex numbers with a negative real part. Consequently, for parameters $(\omega _{0}, k_{\vartheta })$ from regions III and V, the family of fixed points (2) of the reduced system (1) with additional control of the form (11) is of stable node-focus type.

• 3. For parameters (ω ₀,k _𝜗 ) from region VI in Fig. 4 (a narrow wedge-shaped region for large values of $k_{\vartheta }$), all roots of the polynomial (13) are real (negative) numbers. Consequently, the family of fixed points (2) of the reduced system (1) with additional control of the form (11) is of stable node type.