Differential Geometry

Transcription

1 Appendix A Differential Geometry Differential geometry is the study of geometry using the principles of calculus. In general, a curve r(q) is defined as a vector-valued function in R n space. The parameter q varies over a R number line. Mathematically, this is a continuous mapping r : I R n,wherei [a, b] andq I. For example, acurver(q) in 3D is represented as r : I R 3,wherer(q) = (x(q), y(q), z(q)). Thus a curve r(q) can be considered as a position vector in Euclidean space. The function r(q) traces the curve as the parameter q varies. If the parameter q is time, the position vector will be given by a vector from the origin to the curve (x(q), y(q), z(q)) at time q. The velocity and acceleration can simply be calculated by taking the derivative of the curve, and their profiles can be drawn by substituting the values of q. The geometric properties of the curve or path per se can be studied by unit speed parametrisation as follows. The arc length h(q) ofthecurver(q) is s2 h(q) = ẋ 2 + ẏ 2 + ż 2 dq. (A.1) s 1 The unit speed parametrisation is such that the parametric speed ṡ = ds/dq of the path is unity. This is an ideal concept. This is explained as follows. Consider a vehicle that starts moving at time q 1 and then stops at time q 2. Thepathlengthattimeq 1 is h 1 and at time q 2 is h 2. A path of unit speed parametrisation has (q 2 q 1 ) = (h 2 h 1 ). This means that the time travelled Cooperative Path Planning of Unmanned Aerial Vehicles Antonios Tsourdos, Brian White and and Madhavan Shanmugavel 2011 John Wiley & Sons, Ltd. ISBN:

2 176 Appendix A: Differential Geometry is equal to the distance travelled. Mathematically, this is dr dh = dr/dq dh/dq = 1. (A.2) The physical significance of differential geometry of the curve is as follows. Taking q as time, the first derivative is the tangent vector and it defines velocity. The speed is given by the modulus of the velocity vector, and the direction of velocity (heading) is specified by the unit tangent vector, t. The second derivative is the acceleration vector, and this has two components, one along the tangent and other normal to the tangent. The tangential acceleration is given by the second derivative of the velocity vector and its direction is along the direction of the heading velocity. The direction of the normal acceleration is given by a unit normal vector, n, and its magnitude is equal to the centripetal acceleration given by κ v 2,whereκ is the curvature and v is the velocity. Thus the curvature is proportional to the lateral acceleration and hence the lateral force induced while the vehicle is turning. Taking the path length as a parameter, the rate of change of the tangent vector with respect to the arc length defines the tangent vector. The cross-product of the unit vectors t and n produces a third unit vector, called the binormal vector b, which is orthogonal to t and n. Thus the orthogonal triad (t, n, b) forms a moving frame on the curve. The plane spanned by the vectors t and n is the osculating plane. The vectors n and b form the normal plane, and the vectors b and t form the rectifying plane. These three planes are orthogonal to each other. A continuous sequence of this triad represents the orientation of the curve in space. The curvature and torsion (κ and τ) completely specify a path in space. Thus we have: unit tangent vector, t = ṙ(q) ṙ(q), (A.3) unit binormal vector, ṙ(q) r(q) b = ṙ(q) r(q), (A.4) unit normal vector, n = b t. (A.5) The curvature profile at a point P is defined by the relation κ = dɛ dh, (A.6)

3 Appendix A: Differential Geometry 177 where h is the path length and ɛ is the angle subtended by the tangent with the x axis. But, Hence, equation (A.6) becomes dɛ dh = dɛ/dq dh/dq. ω = vκ, (A.7) where ω = dɛ/dq is the angular velocity, v = dh/dq is the linear velocity and q is the path parameter. A.1 Frenet Serret Equations At every point on the curve, we can fix a local frame formed by the tangent, normal and binormal orthonormal vectors. Such a frame is called the Frenet Serret (FS) frame (Figure A.1). The FS equations describe the Z b t e z P r(q) s(q) e y O e x n X e y Y Figure A.1 Frenet Serret frame {t, n, b}, inwhicht is the unit tangent, n is the unit normal and b is the unit binormal. On the diagram, r(q) isthepath,p is the position vector of a point on the path, {e x, e y, e z } are the unit vectors and h(q) is the path length

4 178 Appendix A: Differential Geometry rate of change of the curve with respect to the change of arc length. The FS equations are as follows: t = κ(q)n, n = κ(q)t + τ(q)b, b = τ(q)n. In matrix form, this becomes t 0 κ(q) 0 n = κ(q) 0 τ(q) b 0 τ(q) 0 t n b, (A.8) (A.9) (A.10) (A.11) where the prime represents the derivative with respect to the path variable q and curvature, κ(q) = r (q) r (q), (A.12) torsion, τ(q) = [r (q) r (q) r (q)] κ 2. (A.13) (q) The time rate of change of the FS vectors in matrix form is ṫ 0 κ(t) 0 t ṅ = q κ(t) 0 τ(t) n, ḃ 0 τ(t) 0 b (A.14) where q = dq/dt is the speed (parametric speed) and q is the path parameter. Thus we obtain curvature, κ(t) = torsion, τ(t) = A.2 Importance of Curvature and Torsion ṙ(t) r(t) ṙ(t) 3, (A.15) ṙ(t) r(t)... r (t) ṙ(t) r(t) 2. (A.16) Mathematically, a flyable path is a regular curve that captures both the geometric (locus of points) and kinematic (motion) aspects. A regular curve r is a mapping r :[a, b] R at least three times continuously differentiable, r C 3 and satisfying the regularity condition dr/dq 0 for all q [a, b]. Regularity

5 Appendix A: Differential Geometry 179 means that the point moving along the curve is not allowed to stop, a natural requirement for fixed-wing UAVs. However, considering the kinematic constraints, it is important for the path to have curvature continuity. By the principles of differential geometry (Kreyszig 1991; Lipschutz 1969), the curvature and torsion are fundamental properties of a path, by which a curve is completely determined in space. In two dimensions, curvature alone is enough. Apart from the geometric insights, these two properties play an important role in the mechanics of a moving vehicle. The physical significance of these properties are that the curvature is proportional to the lateral acceleration and is measured by the rate of change of the tangent vector, while the torsion is proportional to the angular momentum and is measured by the rate of change of the tangent plane: κ(q) = ṙ r ṙ 3, τ(q) = (A.17) ṙ, r,... r ṙ r 3. (A.18) From equation (A.18), the curvature and torsion, respectively, are functions of the first two and three derivatives of the path. Hence, it is necessary to have a path of minimal order sufficient to satisfy curvature constraints and additional flexibility to negotiate obstacles. A.3 Motion and Frames The design of the Dubins path using analytic geometry is as simple and easy to understand as the Euclidean space is familiar to us. However, for an autonomous vehicle, it would be appropriate to use frames to describe the motion. A curve can be studied by assigning a frame at each point on it. The curve evolves with the rate of change of these frames (O Neill 1967). The Frenet Serret frame (FS) is one such frame, shown in Figure A.2. This frame constitutes tangent (t), normal (n) and binormal (b) vectors, which together form a trihedron on every point of the path. The advantage of the frame is that the rate of change of the trihedron varies with the frame itself with the given curvatures of the path: t = ṙ ṙ, n = ṙ r, b = t n, (A.19) ṙ r

6 180 Appendix A: Differential Geometry z Q t 2 b 2 n 1 b 1 r (q) y n 2 t 1 q s P x Figure A.2 Frenet Serret frame on a 3D curve where the derivatives are with respect to the path parameter q. Therateof change of these vectors (and hence the frame) is a function of two parameters, curvature and torsion: ṫ ṅ ḃ = v(q) 0 κ 0 κ 0 τ 0 τ 0 t n b, (A.20) where v is the velocity. From equation (A.20), it can be seen that, for a given curvature and torsion, the evolution of the FS frame with time is the frame itself. Kinematics can be best represented using differential geometry. Differential geometry enables the motion along the curve to be understood rather than representing motion with respect to some fixed frame. For example, taking time q as the path parameter, the path r(q) represents the equation of motion of the vehicle along the path with time. Also, the motion can be expressed in terms of a moving trihedron along the curve. This moving trihedron is purely a function of the intrinsic properties of the path, namely (i) curvature and (ii) torsion, as shown in Figure A.2. The Frenet Serret (FS) frame forms a basis at each point on the curve. Hence, the curve can be studied and generated by transformation of these

7 Appendix A: Differential Geometry 181 bases. From Figure A.2, the curve r(q) is generated by transforming the frame F 1 (t 1, n 1, b 1 )atp to a new frame F 2 (t 2, n 2, b 2 )atq by F 2 = R ɛ F 1,whereR ɛ is the rotation matrix with rotation angle ɛ. References Kreyszig, E Differential Geometry. Dover Publications. Lipschutz, M Schaum s Outline of Differential Geometry. McGraw-Hill. O Neill, B Elementary Differential Geometry. Academic Press.