Autonomous Underwater Vehicles: Equations of Motion Monique Chyba - November 18, 2015 Departments of Mathematics, University of Hawai i at Mānoa Elective in Robotics 2015/2016 - Control of Unmanned Vehicles
Literature Classical mechanics text for the equations of motion: Lamb (1961), Ardema (2005) or Meriam & Kraige (1997). The addition of the hydrodynamic forces and moments into these general equations can be found in Lamb (1945) (see also Fossen 1994), with an in-depth treatment of these topics presented in Newman (1977). Ardema, M. D. (2005), Newton-Euler Dynamics, Springer, New York. Fossen, T. I. (1994), Guidance and Control of Ocean Vehicles, John Wiley & Sons. Lamb, H. (1945), Hydrodynamics, 6th edn, Dover Publications. Lamb, H. (1961), Dynamics, University Press, Cambridge. Meriam, J. & Kraige, L. (1997), Engineering Mechanics, DYNAMICS, 4th edn, JohnWiley & Sons, Inc., New York. Newman, J. (1977), Marine Hydrodynamics, MIT Press, Cambridge, MA.
Rigid Body Kinematics We first need a reference frame from which to measure distances and angles. This is done by choosing an Earth-fixed reference frame. For low-speed marine vehicles such as those studied here, the Earths movement has a negligible effect on the dynamics of the vehicle. Thus, the earth-fixed frame may be considered as an inertial frame. To precisely identify the configuration of a rigid body, we need to know the position and orientation of a point on the body with respect to the inertial reference frame. Thus, we will define a reference frame fixed to a chosen point on the body.
Notations Through the development of the differential geometric theory, the SNAME (1950) standard notation will prove to be notationally cumbersome for summations and geometric representations. To this end, we choose an alternate notation to simplify the expressions.
Special Euclidean Group We identify the position and the orientation of a rigid body with an element of SE(3): (b, R). Here b = (b 1, b 2, b 3 ) t R 3 denotes the position vector of the body, and R SO(3) is a rotation matrix describing the orientation of the body. Velocities The translational and angular velocities in the body-fixed frame are denoted by ν = (ν 1, ν 2, ν 3 ) t and Ω = (Ω 1, Ω 2, Ω 3 ) t respectively. External Forces and moments ϕ ν = (ϕ 1, ϕ 2, ϕ 3 ) t and τ Ω = (τ Ω1, τ Ω2, τ Ω3 ) t account for the control forces. We denote σ = (ϕ ν, τ Ω ) t.
Kinematic equations of a rigid body The kinematic equations of a rigid body are give by: ḃ = Rν, Ṙ = R ˆΩ where the operator ˆ : R 3 so(3) is defined by ŷz = y z with so(3) the space of skew-symmetric matrices. In local coordinates, we have: ḃ 1 = ν 1 cos ψ cos θ + ν 2 R 12 + ν 3 R 13, ḃ 2 = ν 1 sin ψ cos θ + ν 2 R 22 + ν 3 R 23, ḃ 3 = ν 1 sin θ + ν 2 cos θ sin φ + ν 3 cos θ cos φ, φ = Ω 1 + Ω 2 sin φ tan θ + Ω 3 cos φ tan θ, θ = Ω 2 cos φ Ω 3 sin φ, ψ = sin φ cos θ Ω 2 + cos φ cos θ Ω 3
Dynamic equations of a rigid body
Simplified We take the origin of the body-fixed frame to be the center of gravity C G. Moreover, we assume the body to have three planes of symmetry with body axes which coincide with the principal axes of inertia. The kinetic energy of the rigid body is given by T body = 1 2 ( v Ω ) t ( mi3 0 0 J ) ( v Ω where m is the mass of the rigid body, I 3 is the 3 3-identity matrix and J is the body inertia matrix. The equations of motion for a rigid body are: ) (1) M ν = Mν Ω J Ω = JΩ Ω + Mν ν (2) where M = mi 3.
Submerged Rigid Body Dynamics We will now examine the additional forces and moments acting on a rigid body which arise due to submersion in a fluid. We first consider the added mass and inertia resulting from the kinetic energy of the fluid caused by accelerations of the body. Next, we examine the damping and dissipation experienced by a body submerged in a viscous fluid. Finally, we consider the terms required to account for the restoring (conservative) forces and moments, which arise from gravity and buoyancy.
Added Mass The added mass is a pressure-induced force due to the inertia of the surrounding fluid and is proportional to the acceleration of the rigid body. Assume three planes of symmetry, as is common for most AUVs and the added mass matrix is diagonal. Note that the matrix Cor B is different from the previous presentation since the fluid does indeed contribute to the Coriolis and centripetal forces. Imlay, F. (1961), The complete expressions for added mass of a rigid body moving in an ideal fluid, Technical Report DTMB 1528, David Taylor Model Basin, Washington D.C.
Viscous Damping Viscous damping and dissipation encountered by marine vehicles are caused by many factors. The major factors include radiation-induced potential damping from forced body oscillations in the presence of a free surface, linear and quadratic skin friction, wave drift damping and vortex shedding. We assume the drag force D ν (ν) and drag momentum D Ω (Ω) matrices are both diagonal. We have: M ν = Mν Ω + D ν (ν)ν + R t ρgvk + ϕ ν J Ω = JΩ Ω + Mν ν + D Ω (Ω)Ω r B R t ρgvk + τ Ω where D ν (.), D Ω (.) represent the drag force and momentum. Allmendinger, E. E. (1990), Submersible Vehicle Design, SNAME.
Restoring Forces and Moments Restoring forces can be viewed as a force (torque) which acts to pull the rigid body back to its original position or orientation. where W = mg is the submerged weight and B = ρgv the buoyancy force.
Under our assumptions, the equations of motion, in the body-fixed frame, for a controlled rigid body submerged in a real fluid are given by: M ν = Mν Ω + D ν (ν)ν + R t ρgv k + ϕ ν J Ω = JΩ Ω + Mν ν + D Ω (Ω)Ω r B R t ρgv k + τ Ω where M accounts for the mass and added mass coefficients, J accounts for the body moments of inertia and added moments of inertia coefficients. The matrices D ν (ν), D Ω (Ω) represent the drag force and momentum. And, ϕ ν = (ϕ ν1, ϕ ν2, ϕ ν3 ) t and τ Ω = (τ Ω1, τ Ω2, τ Ω3 ) t account for the control forces.
Equivalent Expression
Kinetic Energy Metric The kinetic energy metric is the unique Riemannian metric on Q SO(3) defined by ( ) mi3 + M G = f 0 0 J b + J f Bullo, F. & Lewis, A. D. (2005a), Geometric Control of Mechanical Systems, Springer. do Carmo, M. (1992), Riemannian Geometry, Birkhauser, Boston. slides-bullo.pdf
Levi-Civita Connection It is the unique affine connection that is both symmetric and metric compatible. The Levi-Civita connection provides the appropriate notion of acceleration for a curve in the configuration space by guaranteeing that the acceleration is in fact a tangent vector field along a curve. The connection also accounts for the Coriolis and centripetal forces acting on the system. Explicitly, if γ(t) = (b(t), R(t)) is a curve in SE(3), and γ (t) = (ν(t), Ω(t)) is its pseudo-velocity, the acceleration is given by ( γ γ ν + M 1( Ω Mν ) ) = Ω + J 1( Ω JΩ + ν Mν ), (3) where denotes the Levi-Civita connection and γ γ is the covariant derivative of γ with respect to itself.
Submerged rigid-body is a first-order affine control system on the tangent bundle T SE(3) which represents the second-order forced affine-connection control system on SE(3) ( γ γ M = 1( ) ) D ν (ν)ν + ϕ ν J 1( D Ω (Ω)Ω r CB R t ). (4) ρgvk + τ Ω Introducing σ = (ϕ ν, τ Ω ), equation (4) takes the form: γ γ = Y (γ(t)) + 6 i=1 with I 1 i being column i of the matrix I 1 = Y (γ(t)) accounts for the external forces. I 1 i (γ(t))σ i (t) (5) ( ) M 1 0 and 0 J 1