A global steering solution for exact and real-time singularity avoidance can be obtained by constraining the system to operate within the restricted workspace of Fig. 6.11. This can be implemented by applying relationship (6.3) to equation (4.5) and formulating a steering law based on the parameters σ, γ and α. This algebraically constrains gimbal steering by one degree of freedom and imposes the restraining condition in equation (6.4). The solution for the desired torque becomes:
˙
φ∗=−(C∗)−1τu (6.6)
The steering parameter φ∗ = [σ, γ, α]T and its derivative have the same dimensions as the vector of desired torque and so the constrained JacobianC∗ is a [3×3] square matrix with an inverse that can be computed directly. The following condition must be satisfied:
C∗φ˙∗ = Cφ˙¯
The expression for Ccan be decomposed into its direct and coupled parts as follows: (C∗o+C∗×) ˙φ∗ = (Co+C×) ˙¯φ
The first term can be expanded by using (4.4) and substituting in (6.1) and (6.3) to give:
C∗oφ˙∗ = −¯cdiag(Jhψ˙¯) ˙¯φ
= Jhψ˙¯
−c∗cos(α+σ) sin(α+γ) c∗cos(α−σ) −sin(α−γ)
−sin(α+σ) −c∗cos(α+γ) sin(α−σ) c∗cos(α−γ)
−s∗cos(α+σ) −s∗cos(α+γ) −s∗cos(α−σ) −s∗cos(α−γ)
˙ α+ ˙σ ˙ α+ ˙γ ˙ α−σ˙ ˙ α−γ˙ = Jhψ˙¯
−2c∗cosαcosσσ˙ + 2 sinαcosγγ˙ + 2(c∗sinαsinσ+ cosαsinγ) ˙α −2 sinαcosσσ˙ −2c∗cosαcosγγ˙ + 2(c∗sinαsinγ−cosαsinσ) ˙α
2s∗sinαsinσσ˙ + 2s∗sinαsinγγ˙ −2(s∗cosαcosσ+s∗cosαcosγ) ˙α
= −2Jhψ˙¯
c∗cosαcosσ −sinαcosγ −(c∗sinαsinσ+ cosαsinγ)
c∗cosαcosγ sinαcosσ (cosαsinσ−c∗sinαsinγ)
−s∗sinαsinσ −s∗sinαsinγ (s∗cosαcosσ+s∗cosαcosγ)
˙ σ ˙ γ ˙ α
Similarly the second term can be expanded to give:
C∗×φ˙∗= 1 2[(h1c
T
1 +c1hT1)(ω+ωd) ...(hNcTN +cNhTN)(ω+ωd)](Jc−Jh) ˙¯φ= 0
This expression is zero for CMG systems that are symmetrical about the principal axes of the body, since (hicTi +cihTi ) becomes zero. Thus the constrained steering law (6.6) becomes:
˙ φ∗ = 1 2Jhψ˙¯
c∗cosαcosσ −sinαcosγ −(c∗sinαsinσ+ cosαsinγ)
c∗cosαcosγ sinαcosσ (cosαsinσ−c∗sinαsinγ)
−s∗sinαsinσ −s∗sinαsinγ (s∗cosαcosσ+s∗cosαcosγ)
−1
Chapter Six 6.4. Exact and real-time steering
Once the steering parameter rates have been computed the gimbal rate commands can be obtained as follows:
˙¯
φ= [ ˙α+ ˙σ,α˙ + ˙γ,α˙−σ,˙ α˙ −γ˙]T
Similarly the steering parameters in (6.7) can be determined from the measured gimbal angles using the following equations:
α= φ1+φ2+φ3+φ4 4 , σ= φ1−φ3 2 , γ = φ2−φ4 2
In practical applications it is possible that condition (6.4) is violated as the frequency of the control system is finite. In these situations null motion can be used to bring the system back to this condition using the following null gain:
knull=−κ(φ1−φ2+φ3−φ4)
where the sensitivity is tuned by adjusting κ. A schematic of the steering law is illustrated in Fig. 6.12.
This steering law guarantees exact and real-time steering within a restricted workspace. This is a simple solution to the complex problem of singularity avoidance that eliminates the problem of internal singularities by applying algebraic constraints on gimbal motion. The uniqueness of its response is guaranteed by limiting the domain of the three steering param- etersσ,γ, andα to within±π/2, ensuring the repeatability of the system’s performance and limiting the gimbal angles to within ±90o. This is of significant practical importance as it prevents multiple rotations of the gimbals, which is often overlooked in simulations and would require complicated mechanisms for actuation of the flywheels and serve no useful purpose. The steering law not only simplifies the design of the hardware, but also simplifies the imple- mentation in software as the matrix inversion does not require the real-time computation of a complicated pseudo-inverse Jacobian and thus reduces the burden on the processors used in the AUV.
Chapter 7
Development of a Zero-G Class
underwater robot
This chapter follows the design and construction of the CMG actuated Zero-G Class underwa- ter robot IKURA. IKURA is an experimental platform developed as part of this research to verify the associated theoretical developments, assess the practical application of the control system and to demonstrate unrestricted attitude control and three-dimensional manoeuvering capabilities. This is the first application of CMGs to underwater robots and so the practical considerations of the mechanical and electrical design of the system are discussed. This chap- ter is supported by Appendices A - C. Appendix A contains CAD drawings and photographs of IKURA and its CMG system. Appendix B contains details of the electronic design, signals and circuitry. The hydrodynamic parameters are presented together with other modelling parameters in Appendix C.
7.1
Zero-G classification
A Zero-G Class underwater robot is defined as any underwater robot that can:
• adopt and maintain any attitude on the surface of a sphere with a zero radius turning circle and
• actively stabilise any attitude while translating in surge.
It should be noted that these points are not intended to specify the entire design of a complete AUV system, but simply outline the primary requirements for the attitude control capabilities of this new class of underwater robot. To satisfy the first point, the robot should ideally have coincident centres of gravity and buoyancy and thus zero righting moment. Unrestricted attitude control can be provided by the proposed CMG based method. The robot should be capable of fast, on the spot reorientations to offer sufficiently agile manoeuvrability to meet its specification. An example of the specification used in this research is detailed in the next section. The second point can be satisfied by equipping the robot with a single thruster to actuate surge. This is not restrictive as missions can be approached in a fully three-dimensional manner by making use of the attitude control capabilities.