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non-holonomic constraints of rigid bodies, he introduced the form with multipliers, which is now called the
Lagrange equations of the second kind with multipliers. The terms the holonomic and nonholonomic systems were introduced by Heinrich Hertz in 1894. In 1897, S. A. Chaplygin first suggested to form the equations of motion without Lagrange multipliers. Under certain linear constraints, he introduced on the left-hand side of the equations of motion a group of extra terms of the Lagrange-operator type. The remaining extra terms characterise the nonholonomicity of system and they become zero when the given constrains are integrable. In 1901 P. V.Voronets generalised Chaplygin's work to the cases of noncyclic holonomic coordinates and of nonstationary constraints.
3574:. In the local coordinate frame the pendulum is swinging in a vertical plane with a particular orientation with respect to geographic north at the outset of the path. The implicit trajectory of the system is the line of latitude on the Earth where the pendulum is located. Even though the pendulum is stationary in the Earth frame, it is moving in a frame referred to the Sun and rotating in synchrony with the Earth's rate of revolution, so that the only apparent motion of the pendulum plane is that caused by the rotation of the Earth. This latter frame is considered to be an inertial reference frame, although it too is non-inertial in more subtle ways. The Earth frame is well known to be non-inertial, a fact made perceivable by the apparent presence of
96:
modeled by the lower-dimensional space. In contrast, if the system intrinsically cannot be represented by independent coordinates (parameters), then it is truly an anholonomic system. Some authors make much of this by creating a distinction between so-called internal and external states of the system, but in truth, all parameters are necessary to characterize the system, be they representative of "internal" or "external" processes, so the distinction is in fact artificial. However, there is a very real and irreconcilable difference between physical systems that obey conservation principles and those that do not. In the case of
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3408:, a system that is modellable by a Pfaffian constraint must be holonomic if the configuration space consists of two or fewer variables. By modifying our original system to restrict it to have only two degrees of freedom and thus requiring only two variables to be described, and assuming it can be described in Pfaffian form (which in this example we already know is true), we are assured that it is holonomic.
3627:
pitch and radius of the helix. This system is also nonholonomic, for we can easily coil the fiber down in a second helix and align the ends, returning the light to its point of origin. The anholonomy is therefore represented by the deviation of the angle of polarization with each circuit of the fiber. By suitable adjustment of the parameters, it is clear that any possible angular state can be produced.
2143:{\displaystyle A_{\gamma }\left({\frac {\partial A_{\beta }}{\partial u_{\alpha }}}-{\frac {\partial A_{\alpha }}{\partial u_{\beta }}}\right)+A_{\beta }\left({\frac {\partial A_{\alpha }}{\partial u_{\gamma }}}-{\frac {\partial A_{\gamma }}{\partial u_{\alpha }}}\right)+A_{\alpha }\left({\frac {\partial A_{\gamma }}{\partial u_{\beta }}}-{\frac {\partial A_{\beta }}{\partial u_{\gamma }}}\right)=0}
1738:{\displaystyle {\begin{pmatrix}1&0&0&-r\cos \theta \\0&1&0&-r\sin \theta \end{pmatrix}}{\begin{pmatrix}{\dot {x}}\\{\dot {y}}\\{\dot {\theta }}\\{\dot {\phi }}\end{pmatrix}}=\mathbf {0} ={\begin{pmatrix}1&0&0&-r\cos \theta \\0&1&0&-r\sin \theta \end{pmatrix}}{\begin{pmatrix}{\text{d}}x\\{\text{d}}y\\{\text{d}}\theta \\{\text{d}}\phi \end{pmatrix}}}
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the mechanism first incrementing 3 units on the x-axis and then 3 units on the y-axis, incrementing the Y-axis position first, or operating any other sequence of position-changes that result in a final position of 3,3. Since the final state of the machine is the same regardless of the path taken by the plotter-pen to get to its new position, the end result can be said not to be
112:. For parallel transport on a sphere, the implicit dependence is intrinsic to the non-euclidean metric. The surface of a sphere is a two-dimensional space. By raising the dimension, we can more clearly see the nature of the metric, but it is still fundamentally a two-dimensional space with parameters irretrievably entwined in dependency by the
2714:{\displaystyle -r\cos \theta \left({\frac {\partial }{\partial x}}(0)-{\frac {\partial }{\partial \theta }}(1)\right)+0\left({\frac {\partial }{\partial \phi }}(1)-{\frac {\partial }{\partial x}}(-r\cos \theta )\right)+1\left({\frac {\partial }{\partial \theta }}(-r\cos \theta )-{\frac {\partial }{\partial \phi }}(0)\right)=0}
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the same place, the valve will almost certainly not be in the same position as before. Its new position depends on the path taken. If the wheel were holonomic, then the valve stem would always end up in the same position as long as the wheel were always rolled back to the same location on the Earth.
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system where the state of the system's mechanical components will have a single fixed configuration for any given position of the plotter pen. If the pen relocates between positions 0,0 and 3,3, the mechanism's gears will have the same final positions regardless of whether the relocation happens by
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The general character of anholonomic systems is that of implicitly dependent parameters. If the implicit dependency can be removed, for example by raising the dimension of the space, thereby adding at least one additional parameter, the system is not truly nonholonomic, but is simply incompletely
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When linearly polarized light is again introduced at one end, with the orientation of the polarization aligned with the stripe, it will, in general, emerge as linear polarized light aligned not with the stripe, but at some fixed angle to the stripe, dependent upon the length of the fiber, and the
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system, is one in which there is a continuous closed circuit of the governing parameters, by which the system may be transformed from any given state to any other state. Because the final state of the system depends on the intermediate values of its trajectory through parameter space, the system
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Take a length of optical fiber, say three meters, and lay it out in an absolutely straight line. When a vertically polarized beam is introduced at one end, it emerges from the other end, still polarized in the vertical direction. Mark the top of the fiber with a stripe, corresponding with the
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as can, for example, the inverse square law of the gravitational force. This latter is an example of a holonomic system: path integrals in the system depend only upon the initial and final states of the system (positions in the potential), completely independent of the trajectory of transition
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subtended by that circle of latitude. The path need not be constrained to latitude circles. For example, the pendulum might be mounted in an airplane. The anholonomy is still proportional to the solid angle subtended by the path, which may now be quite irregular. The
Foucault pendulum is a
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first suggested to extend the equations of motion with nonholonomic constraints in 1871. He introduced the expressions for
Cartesian velocities in terms of generalized velocities. In 1877, E. Routh wrote the equations with the Lagrange multipliers. In the third edition of his book for linear
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above where it is said, " new position depends on the path taken. If the wheel were holonomic, then the valve stem would always end up in the same position as long as the wheel were always rolled back to the same location on the Earth. Clearly, however, this is not the case, so the system is
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were zero, that that part of the test equation would be trivial to solve and would be equal to zero. Therefore, it is often best practice to have the first test equation have as many non-zero terms as possible to maximize the chance of the sum of them not equaling zero. Therefore, we choose:
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We must now relate these variables to each other. We notice that as the wheel changes its rotation, it changes its position. The change in rotation and position implying velocities must be present, we attempt to relate angular velocity and steering angle to linear velocities by taking simple
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is not actually necessary in the real world as the orientation of the coordinate system itself is arbitrary. The system can become holonomic if the wheel moves only in a straight line at any fixed angle relative to a given reference. Thus, we have not only proved that the original system is
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It is not necessary for all non-holonomic constraints to take this form, in fact it may involve higher derivatives or inequalities. A classical example of an inequality constraint is that of a particle placed on the surface of a sphere, yet is allowed to fall off it:
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54:(the parameters varying continuously in values) but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state. Nonholonomic mechanics is autonomous division of
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Motion along the line of latitude is parameterized by the passage of time, and the
Foucault pendulum's plane of oscillation appears to rotate about the local vertical axis as time passes. The angle of rotation of this plane at a time
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We have completed our proof that the system is nonholonomic, but our test equation gave us some insights about whether the system, if further constrained, could be holonomic. Many times test equations will return a result like
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plotter, the process of moving the pen from 0,0 to 3,3 can result in the gears of the robot's mechanism finishing in different positions depending on the path taken to move between the two positions. See this very similar
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First, we define the configuration space. The wheel can change its state in three ways: having a different rotation about its axle, having a different steering angle, and being at a different location. We may say that
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and therefore one of those two coordinates is completely redundant. We already know that the steering angle is a constant, so that means the holonomic system here needs to only have a configuration space of
3623:. As such the result is a gradual rotation of the fiber about the fiber's axis as the fiber's centerline progresses along the helix. Correspondingly, the stripe also twists about the axis of the helix.
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the failures to gather all requirements to make the system holonomic, if possible. In this system, out of the seven additional test equations, an additional case presents itself:
1844:. If this system were holonomic, we might have to do up to eight tests. However, we can use mathematical intuition to try our best to prove that the system is nonholonomic on the
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Chaplygin, S.A. (1897). "О движении тяжелого тела вращения по горизонтальнойплоскости" [A motion of heavy body of revolution on a horizontal plane].
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equal to zero. This implies that if the wheel were not allowed to turn and had to move only in a straight line at all times, it would be a holonomic system.
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axis. In fact, by selection of a suitable path, the sphere may be re-oriented from the initial orientation to any possible orientation of the sphere with
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1220:{\displaystyle {\begin{pmatrix}{\dot {x}}\\{\dot {y}}\end{pmatrix}}={\begin{pmatrix}r{\dot {\phi }}\cos \theta \\r{\dot {\phi }}\sin \theta \end{pmatrix}}}
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with respect to the initial orientation is the anholonomy of the system. The anholonomy induced by a complete circuit of latitude is proportional to the
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Then, let's separate the variables from their coefficients (left side of equation, derived from above). We also realize that we can multiply all terms by
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Consider a three-dimensional orthogonal
Cartesian coordinate frame, for example, a level table top with a point marked on it for the origin, and the
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nonholonomic." However it is easy to visualize that if the wheel were only allowed to roll in a perfectly straight line and back, the valve stem
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4003:
The
Nonholonomy of the Rolling Sphere, Brody Dylan Johnson, The American Mathematical Monthly, June–July 2007, vol. 114, pp. 500–508.
1387:{\displaystyle {\begin{pmatrix}{\dot {x}}-r{\dot {\phi }}\cos \theta \\{\dot {y}}-r{\dot {\phi }}\sin \theta \end{pmatrix}}=\mathbf {0} }
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It is possible to model the wheel mathematically with a system of constraint equations, and then prove that that system is nonholonomic.
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implying the system could never be constrained to be holonomic without radically altering the system, but in our result we can see that
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in blue. Corresponding to this point is a diameter of the sphere, and the plane orthogonal to this diameter positioned at the center
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There is one thing that we have not yet considered however, that to find all such modifications for a system, one must perform
2879:, the radius of the wheel, can be zero. This is not helpful as the system in practice would lose all of its degrees of freedom.
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Voronets, P. (1901). "Об уравнениях движения для неголономных систем" [Equations of motion of nonholonomic systems].
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subject to differential constraints and non-linear constraints, such that when the system evolves along a path in its
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Now, coil the fiber tightly around a cylinder ten centimeters in diameter. The path of the fiber now describes a
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axes laid out with pencil lines. Take a sphere of unit radius, for example, a ping-pong ball, and mark one point
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nonholonomic, but we also were able to find a restriction that can be added to the system to make it holonomic.
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3542: = 1. The system is therefore nonholonomic. The anholonomy may be represented by the doubly unique
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direction is equal to the angular velocity times the radius times the cosine of the steering angle, and the
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with respect to a given reference frame. In classical mechanics, any constraint that is not expressible as
4015:, Jean-Paul Laumond (Ed.), 1998, Lecture Notes in Control and Information Sciences, Volume 229, Springer,
3745:. Contemporary Mathematics. Vol. 395. Providence, RI: American Mathematical Society. pp. 29–38.
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In order for the above form to be nonholonomic, it is also required that the left hand side neither be a
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1085:{\displaystyle \mathbf {u} ={\begin{bmatrix}x&y&\theta &\phi \end{bmatrix}}^{\mathrm {T} }}
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whose state depends on the path taken in order to achieve it. Such a system is described by a set of
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An individual riding a motorized unicycle. The configuration space of the unicycle, and the radius
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Bryant, Robert L. (2006). "Geometry of manifolds with special holonomy: '100 years of holonomy'".
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Consider the wheel of a bicycle that is parked in a certain place (on the ground). Initially the
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is at a certain position on the wheel. If the bicycle is ridden around, and then parked in
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3919:. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing.
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A wheel (sometimes visualized as a unicycle or a rolling coin) is a nonholonomic system.
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velocity is similar. Now we do some algebraic manipulation to transform the equation to
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3397:{\displaystyle \mathbf {u} ={\begin{bmatrix}x&\phi \end{bmatrix}}^{\mathrm {T} }}
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This does not pose much difficulty, however, as adding the equations and dividing by
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336:{\displaystyle f(\mathbf {r} _{1},\mathbf {r} _{2},\mathbf {r} _{3},\ldots ,t)=0,}
491:{\displaystyle \sum _{i=1}^{n}a_{s,i}\,dq_{i}+a_{s,t}\,dt=0~~~~(s=1,2,\ldots ,k)}
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produced by the specific path under consideration. This term was introduced by
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3882:(IX). отделения физических наук общества любителей естествознания: 10–16.
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3554:) which, when applied to the points that represent the sphere, carries points
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of the sphere defines a great circle called the equator associated with point
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This example is an extension of the 'rolling wheel' problem considered above.
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However, there is something mathematically special about the restriction of
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end up in the same position! In fact, moving parallel to the given angle of
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We can easily see that this system, as described, is nonholonomic, because
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739:{\displaystyle \sum _{i=1}^{n}a_{s,i}\delta q_{i}=0~~~~(s=1,2,\ldots ,k).}
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example for a mathematical explanation of why such a system is holonomic.
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Advanced part of a
Treatise on the Dynamics of a System of Rigid Bodies
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Clearly, however, this is not the case, so the system is nonholonomic.
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3983:(3rd ed.). United States of America: Addison Wesley. p. 16.
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eight test equations (four from each constraint equation) and collect
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The sphere may now be rolled along any continuous closed path in the
888:
Holonomic constraints § Universal test for holonomic constraints
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Berry, Michael (December 1990). "Anticipations of
Geometric Phase".
349:. In other words, a nonholonomic constraint is nonintegrable and in
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axis. This is the initial or reference orientation of the sphere.
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so we end up with only the differentials (right side of equation):
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of the wheel, are marked. The red and blue lines lay on the ground.
3619:. The helix also has the interesting property of having constant
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3126:{\textstyle \theta ={\frac {\pi }{4}}+n\pi ;\;n\in \mathbb {Z} \;}
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so it is possible to test whether it is holonomic, starting with:
891:
3717:
Mechanics of Non-holonomic
Systems A New Class of Control Systems
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in a
Cartesian grid. Combining the two equations and eliminating
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define the spatial position. Thus, the configuration space is:
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150 years of mathematics at Washington University in St. Louis
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is the distance of the particle from the centre of the sphere.
3639:, nonholonomic has been particularly studied in the scope of
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ie Prinzipien derMechanik in neuem Zusammenhange dargestellt
3720:. Vol. 43. Springer Berlin Heidelberg. pp. XXIII.
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Ferrers, N.M. (1872). "Extension of Lagrange's equations".
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Soltakhanov Yushkov Zegzhda, Sh.Kh Mikhail S. (May 2009).
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between those states. The system is therefore said to be
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which with some simple algebraic manipulation becomes:
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An additional example of a nonholonomic system is the
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3942:"Non Holonomic Constraints in Newtonian Mechanics"
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3949:Pedagogical Review from the Classics of Physics
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3844:: CS1 maint: location missing publisher (
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3603:Linear polarized light in an optical fiber
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884:Holonomic constraints § Pfaffian form
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100:on a sphere, the distinction is clear: a
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1848:test. Considering the test equation is:
1842:universal test for holonomic constraints
880:Holonomic constraints § Terminology
71:cannot be represented by a conservative
3915:Torby, Bruce (1984). "Energy Methods".
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2464:We substitute into our test equation:
964:is the steering angle relative to the
542:is the number of constraint equations.
108:fundamentally distinct from that of a
3615:which, like the circle, has constant
3522:no longer extends along the positive
119:By contrast, one can consider an X-Y
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3292:{\displaystyle \theta =\arctan(y/x)}
2152:we can see that if any of the terms
247:{\displaystyle i\in \{1,\ldots ,N\}}
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2417:{\displaystyle u_{\beta }=d\theta }
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4013:Robot Motion Planning and Control
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203:{\displaystyle \mathbf {r} _{i}}
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3951:. stardrive.org. Archived from
3917:Advanced Dynamics for Engineers
3666:Bicycle and motorcycle dynamics
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1:
2846:{\displaystyle r\sin \theta }
2786:is not always equal to zero.
2263:{\displaystyle A_{\alpha }=1}
520:is the number of coordinates.
3940:Jack Sarfatti (2000-03-26).
2900:{\displaystyle \sin \theta }
2779:{\displaystyle \sin \theta }
2297:{\displaystyle A_{\beta }=0}
839:is the radius of the sphere.
27:Type of optimization problem
2226:{\displaystyle A_{\gamma }}
2172:{\displaystyle A_{\alpha }}
1414:{\displaystyle {\text{d}}t}
856:
4070:
2199:{\displaystyle A_{\beta }}
877:
3876:антpопологии и этногpафии
181:particles with positions
3681:Parallel parking problem
3562:to their new positions.
874:Mathematical explanation
3691:Udwadia–Kalaba equation
3676:Goryachev–Chaplygin top
3312:{\displaystyle \theta }
3075:which has the solution
2920:{\displaystyle \theta }
2907:can be zero by setting
957:{\displaystyle \theta }
857:Layperson's explanation
599:{\displaystyle a_{s,i}}
3752:10.1090/conm/395/07414
3645:feedback linearization
3398:
3339:
3313:
3293:
3247:
3205:
3204:{\displaystyle \pi /4}
3165:
3127:
3067:
3035:
2991:
2971:
2921:
2901:
2873:
2847:
2818:
2790:Additional conclusions
2780:
2752:
2715:
2455:
2418:
2381:
2344:
2298:
2264:
2227:
2200:
2173:
2144:
1832:
1776:
1739:
1415:
1388:
1261:
1241:
1221:
1086:
1018:
998:
978:
958:
938:
917:
910:
833:
811:
789:
740:
652:
600:
565:
536:
514:
492:
380:
337:
248:
204:
175:
4049:Differential topology
4044:Differential geometry
3811:Q. J. Pure Appl. Math
3483: = 1, i.e.
3399:
3340:
3319:, we indeed see that
3314:
3294:
3248:
3206:
3166:
3128:
3068:
3036:
2992:
2972:
2922:
2902:
2874:
2848:
2819:
2781:
2753:
2716:
2456:
2419:
2382:
2345:
2299:
2265:
2228:
2201:
2174:
2145:
1833:
1756:
1740:
1416:
1389:
1262:
1242:
1222:
1087:
1019:
999:
979:
959:
939:
937:{\displaystyle \phi }
911:
895:
834:
812:
790:
741:
632:
623:virtual displacements
601:
566:
564:{\displaystyle q_{i}}
537:
515:
493:
360:
338:
249:
205:
176:
161:Consider a system of
132:. If we substitute a
3901:(22): 659–686.
3696:Lie group integrator
3661:Holonomic constraint
3595:physical example of
3479: = 0, and
3467: = 1, and
3350:
3323:
3303:
3257:
3219:
3187:
3176:layman's explanation
3137:
3079:
3045:
3001:
2981:
2943:
2911:
2885:
2863:
2828:
2817:{\displaystyle -1=0}
2799:
2764:
2727:
2468:
2429:
2392:
2355:
2309:
2275:
2241:
2210:
2183:
2156:
1852:
1753:
1425:
1400:
1275:
1251:
1231:
1227:The velocity in the
1099:
1028:
1008:
988:
968:
948:
928:
900:
823:
801:
753:
629:
577:
548:
526:
504:
357:
347:holonomic constraint
258:
214:
185:
165:
3981:Classical Mechanics
3788:1990PhT....43l..34B
3686:Pfaffian constraint
3671:Falling cat problem
3338:{\displaystyle y=x}
123:as an example of a
102:Riemannian manifold
56:Newtonian mechanics
32:nonholonomic system
4039:Algebraic topology
4021:10.1007/BFb0036069
3977:Goldstein, Herbert
3859:Hertz, H. (1894).
3828:Routh, E. (1884).
3597:parallel transport
3576:centrifugal forces
3394:
3380:
3335:
3309:
3289:
3243:
3201:
3174:Refer back to the
3161:
3123:
3063:
3031:
2987:
2967:
2917:
2897:
2869:
2843:
2814:
2776:
2748:
2711:
2451:
2414:
2377:
2340:
2294:
2260:
2223:
2196:
2169:
2140:
1828:
1735:
1729:
1669:
1578:
1502:
1411:
1384:
1370:
1257:
1237:
1217:
1211:
1140:
1082:
1068:
1014:
994:
974:
954:
934:
918:
906:
829:
807:
785:
736:
616:integrating factor
612:total differential
596:
561:
532:
510:
488:
333:
244:
200:
171:
98:parallel transport
73:potential function
4054:Dynamical systems
3572:Foucault pendulum
3566:Foucault pendulum
3096:
2990:{\displaystyle r}
2872:{\displaystyle r}
2689:
2650:
2598:
2571:
2531:
2504:
2127:
2090:
2033:
1996:
1939:
1902:
1722:
1710:
1698:
1686:
1573:
1557:
1541:
1525:
1406:
1356:
1338:
1313:
1295:
1260:{\displaystyle y}
1240:{\displaystyle x}
1197:
1169:
1135:
1119:
1017:{\displaystyle y}
997:{\displaystyle x}
977:{\displaystyle x}
909:{\displaystyle r}
832:{\displaystyle a}
810:{\displaystyle r}
699:
696:
693:
690:
606:are coefficients.
535:{\displaystyle k}
513:{\displaystyle n}
454:
451:
448:
445:
174:{\displaystyle N}
114:Riemannian metric
16:(Redirected from
4061:
4024:
4010:
4004:
4001:
3995:
3994:
3973:
3967:
3966:
3964:
3963:
3957:
3946:
3937:
3931:
3930:
3912:
3903:
3902:
3890:
3884:
3883:
3871:
3865:
3864:
3856:
3850:
3849:
3843:
3835:
3825:
3819:
3818:
3806:
3800:
3799:
3796:10.1063/1.881219
3771:
3765:
3764:
3754:
3738:
3732:
3731:
3711:
3538: = 0,
3534: = 0,
3510: = 0,
3506: = 0,
3475: = 1,
3463: = 0,
3459: = 0,
3403:
3401:
3400:
3395:
3393:
3392:
3391:
3385:
3384:
3357:
3344:
3342:
3341:
3336:
3318:
3316:
3315:
3310:
3298:
3296:
3295:
3290:
3282:
3252:
3250:
3249:
3244:
3210:
3208:
3207:
3202:
3197:
3170:
3168:
3167:
3162:
3132:
3130:
3129:
3124:
3121:
3097:
3089:
3072:
3070:
3069:
3064:
3040:
3038:
3037:
3032:
2996:
2994:
2993:
2988:
2976:
2974:
2973:
2968:
2926:
2924:
2923:
2918:
2906:
2904:
2903:
2898:
2878:
2876:
2875:
2870:
2852:
2850:
2849:
2844:
2823:
2821:
2820:
2815:
2785:
2783:
2782:
2777:
2757:
2755:
2754:
2749:
2720:
2718:
2717:
2712:
2704:
2700:
2690:
2688:
2677:
2651:
2649:
2638:
2625:
2621:
2599:
2597:
2586:
2572:
2570:
2559:
2546:
2542:
2532:
2530:
2519:
2505:
2503:
2492:
2460:
2458:
2457:
2452:
2441:
2440:
2423:
2421:
2420:
2415:
2404:
2403:
2386:
2384:
2383:
2378:
2367:
2366:
2349:
2347:
2346:
2341:
2321:
2320:
2303:
2301:
2300:
2295:
2287:
2286:
2269:
2267:
2266:
2261:
2253:
2252:
2232:
2230:
2229:
2224:
2222:
2221:
2205:
2203:
2202:
2197:
2195:
2194:
2178:
2176:
2175:
2170:
2168:
2167:
2149:
2147:
2146:
2141:
2133:
2129:
2128:
2126:
2125:
2124:
2111:
2110:
2109:
2096:
2091:
2089:
2088:
2087:
2074:
2073:
2072:
2059:
2052:
2051:
2039:
2035:
2034:
2032:
2031:
2030:
2017:
2016:
2015:
2002:
1997:
1995:
1994:
1993:
1980:
1979:
1978:
1965:
1958:
1957:
1945:
1941:
1940:
1938:
1937:
1936:
1923:
1922:
1921:
1908:
1903:
1901:
1900:
1899:
1886:
1885:
1884:
1871:
1864:
1863:
1837:
1835:
1834:
1829:
1802:
1801:
1789:
1788:
1775:
1770:
1744:
1742:
1741:
1736:
1734:
1733:
1723:
1720:
1711:
1708:
1699:
1696:
1687:
1684:
1674:
1673:
1591:
1583:
1582:
1575:
1574:
1566:
1559:
1558:
1550:
1543:
1542:
1534:
1527:
1526:
1518:
1507:
1506:
1420:
1418:
1417:
1412:
1407:
1404:
1393:
1391:
1390:
1385:
1383:
1375:
1374:
1358:
1357:
1349:
1340:
1339:
1331:
1315:
1314:
1306:
1297:
1296:
1288:
1266:
1264:
1263:
1258:
1246:
1244:
1243:
1238:
1226:
1224:
1223:
1218:
1216:
1215:
1199:
1198:
1190:
1171:
1170:
1162:
1145:
1144:
1137:
1136:
1128:
1121:
1120:
1112:
1091:
1089:
1088:
1083:
1081:
1080:
1079:
1073:
1072:
1035:
1023:
1021:
1020:
1015:
1003:
1001:
1000:
995:
983:
981:
980:
975:
963:
961:
960:
955:
943:
941:
940:
935:
915:
913:
912:
907:
838:
836:
835:
830:
816:
814:
813:
808:
794:
792:
791:
786:
778:
777:
765:
764:
745:
743:
742:
737:
697:
694:
691:
688:
681:
680:
668:
667:
651:
646:
605:
603:
602:
597:
595:
594:
571:are coordinates.
570:
568:
567:
562:
560:
559:
541:
539:
538:
533:
519:
517:
516:
511:
497:
495:
494:
489:
452:
449:
446:
443:
429:
428:
410:
409:
396:
395:
379:
374:
342:
340:
339:
334:
308:
307:
302:
293:
292:
287:
278:
277:
272:
253:
251:
250:
245:
209:
207:
206:
201:
199:
198:
193:
180:
178:
177:
172:
21:
4069:
4068:
4064:
4063:
4062:
4060:
4059:
4058:
4029:
4028:
4027:
4011:
4007:
4002:
3998:
3991:
3975:
3974:
3970:
3961:
3959:
3955:
3944:
3939:
3938:
3934:
3927:
3914:
3913:
3906:
3892:
3891:
3887:
3873:
3872:
3868:
3858:
3857:
3853:
3836:
3827:
3826:
3822:
3808:
3807:
3803:
3773:
3772:
3768:
3740:
3739:
3735:
3728:
3713:
3712:
3708:
3704:
3657:
3641:motion planning
3633:
3605:
3568:
3414:
3404:. As discussed
3379:
3378:
3373:
3363:
3361:
3348:
3347:
3321:
3320:
3301:
3300:
3255:
3254:
3217:
3216:
3185:
3184:
3135:
3134:
3077:
3076:
3043:
3042:
2999:
2998:
2979:
2978:
2941:
2940:
2909:
2908:
2883:
2882:
2861:
2860:
2826:
2825:
2797:
2796:
2792:
2762:
2761:
2725:
2724:
2681:
2642:
2636:
2632:
2590:
2563:
2557:
2553:
2523:
2496:
2490:
2486:
2466:
2465:
2432:
2427:
2426:
2395:
2390:
2389:
2358:
2353:
2352:
2312:
2307:
2306:
2278:
2273:
2272:
2244:
2239:
2238:
2213:
2208:
2207:
2186:
2181:
2180:
2159:
2154:
2153:
2116:
2112:
2101:
2097:
2079:
2075:
2064:
2060:
2057:
2053:
2043:
2022:
2018:
2007:
2003:
1985:
1981:
1970:
1966:
1963:
1959:
1949:
1928:
1924:
1913:
1909:
1891:
1887:
1876:
1872:
1869:
1865:
1855:
1850:
1849:
1840:We now use the
1793:
1777:
1751:
1750:
1728:
1727:
1716:
1715:
1704:
1703:
1692:
1691:
1676:
1668:
1667:
1650:
1645:
1640:
1634:
1633:
1616:
1611:
1606:
1596:
1577:
1576:
1561:
1560:
1545:
1544:
1529:
1528:
1509:
1501:
1500:
1483:
1478:
1473:
1467:
1466:
1449:
1444:
1439:
1429:
1423:
1422:
1398:
1397:
1369:
1368:
1326:
1325:
1279:
1273:
1272:
1249:
1248:
1229:
1228:
1210:
1209:
1182:
1181:
1150:
1139:
1138:
1123:
1122:
1103:
1097:
1096:
1067:
1066:
1061:
1056:
1051:
1041:
1039:
1026:
1025:
1006:
1005:
986:
985:
966:
965:
946:
945:
926:
925:
898:
897:
890:
876:
863:inflation valve
859:
851:
846:
821:
820:
799:
798:
769:
756:
751:
750:
672:
653:
627:
626:
580:
575:
574:
551:
546:
545:
524:
523:
502:
501:
414:
401:
381:
355:
354:
297:
282:
267:
256:
255:
212:
211:
188:
183:
182:
163:
162:
159:
147:
110:Euclidean space
64:
52:parameter space
44:physical system
28:
23:
22:
15:
12:
11:
5:
4067:
4065:
4057:
4056:
4051:
4046:
4041:
4031:
4030:
4026:
4025:
4005:
3996:
3989:
3968:
3932:
3925:
3904:
3897:(in Russian).
3885:
3878:(in Russian).
3866:
3851:
3820:
3801:
3766:
3733:
3726:
3705:
3703:
3700:
3699:
3698:
3693:
3688:
3683:
3678:
3673:
3668:
3663:
3656:
3653:
3632:
3629:
3604:
3601:
3567:
3564:
3471:is located at
3455:is located at
3413:
3412:Rolling sphere
3410:
3390:
3383:
3377:
3374:
3372:
3369:
3368:
3366:
3360:
3356:
3334:
3331:
3328:
3308:
3288:
3285:
3281:
3277:
3274:
3271:
3268:
3265:
3262:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3200:
3196:
3192:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3120:
3116:
3113:
3109:
3106:
3103:
3100:
3095:
3092:
3087:
3084:
3062:
3059:
3056:
3053:
3050:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
2986:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2929:
2928:
2916:
2896:
2893:
2890:
2880:
2868:
2842:
2839:
2836:
2833:
2813:
2810:
2807:
2804:
2791:
2788:
2775:
2772:
2769:
2747:
2744:
2741:
2738:
2735:
2732:
2723:and simplify:
2710:
2707:
2703:
2699:
2696:
2693:
2687:
2684:
2680:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2648:
2645:
2641:
2635:
2631:
2628:
2624:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2596:
2593:
2589:
2584:
2581:
2578:
2575:
2569:
2566:
2562:
2556:
2552:
2549:
2545:
2541:
2538:
2535:
2529:
2526:
2522:
2517:
2514:
2511:
2508:
2502:
2499:
2495:
2489:
2485:
2482:
2479:
2476:
2473:
2462:
2461:
2450:
2447:
2444:
2439:
2435:
2424:
2413:
2410:
2407:
2402:
2398:
2387:
2376:
2373:
2370:
2365:
2361:
2350:
2339:
2336:
2333:
2330:
2327:
2324:
2319:
2315:
2304:
2293:
2290:
2285:
2281:
2270:
2259:
2256:
2251:
2247:
2220:
2216:
2193:
2189:
2166:
2162:
2139:
2136:
2132:
2123:
2119:
2115:
2108:
2104:
2100:
2094:
2086:
2082:
2078:
2071:
2067:
2063:
2056:
2050:
2046:
2042:
2038:
2029:
2025:
2021:
2014:
2010:
2006:
2000:
1992:
1988:
1984:
1977:
1973:
1969:
1962:
1956:
1952:
1948:
1944:
1935:
1931:
1927:
1920:
1916:
1912:
1906:
1898:
1894:
1890:
1883:
1879:
1875:
1868:
1862:
1858:
1827:
1824:
1821:
1818:
1815:
1811:
1808:
1805:
1800:
1796:
1792:
1787:
1784:
1780:
1774:
1769:
1766:
1763:
1759:
1732:
1726:
1718:
1717:
1714:
1706:
1705:
1702:
1694:
1693:
1690:
1682:
1681:
1679:
1672:
1666:
1663:
1660:
1657:
1654:
1651:
1649:
1646:
1644:
1641:
1639:
1636:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
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130:path-dependent
90:Heinrich Hertz
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26:
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14:
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2:
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3992:
3990:0-201-65702-3
3986:
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3978:
3972:
3969:
3958:on 2007-10-20
3954:
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3926:0-03-063366-4
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3782:(12): 34–40.
3781:
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3776:Physics Today
3770:
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3649:mobile robots
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2191:
2187:
2164:
2160:
2150:
2137:
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2106:
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2092:
2084:
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2023:
2012:
2008:
1998:
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1794:
1790:
1785:
1782:
1778:
1772:
1767:
1764:
1761:
1757:
1748:
1747:Pfaffian form
1730:
1724:
1712:
1700:
1688:
1677:
1670:
1664:
1661:
1658:
1655:
1652:
1647:
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1408:
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1371:
1365:
1362:
1359:
1353:
1350:
1344:
1341:
1335:
1332:
1322:
1319:
1316:
1310:
1307:
1301:
1298:
1292:
1289:
1280:
1270:
1269:Pfaffian form
1254:
1234:
1212:
1206:
1203:
1200:
1194:
1191:
1185:
1178:
1175:
1172:
1166:
1163:
1157:
1151:
1146:
1141:
1132:
1129:
1116:
1113:
1104:
1092:
1069:
1063:
1058:
1053:
1048:
1042:
1036:
1011:
991:
971:
951:
931:
921:
903:
894:
889:
885:
881:
873:
871:
868:
864:
854:
849:Rolling wheel
848:
843:
826:
819:
804:
797:
796:
795:
782:
779:
774:
770:
766:
761:
757:
746:
733:
727:
724:
721:
718:
715:
712:
709:
706:
703:
685:
682:
677:
673:
669:
664:
661:
658:
654:
648:
643:
640:
637:
633:
624:
619:
617:
613:
591:
588:
585:
581:
573:
556:
552:
544:
529:
522:
507:
500:
499:
498:
482:
479:
476:
473:
470:
467:
464:
461:
458:
440:
437:
434:
431:
425:
422:
419:
415:
411:
406:
402:
398:
392:
389:
386:
382:
376:
371:
368:
365:
361:
352:
351:Pfaffian form
348:
343:
330:
327:
324:
318:
315:
312:
309:
304:
294:
289:
279:
274:
261:
238:
235:
232:
229:
226:
220:
217:
195:
168:
156:
154:
151:
150:N. M. Ferrers
144:
142:
140:
135:
131:
126:
122:
117:
115:
111:
107:
103:
99:
93:
91:
87:
83:
82:nonintegrable
79:
74:
69:
61:
59:
57:
53:
49:
45:
41:
37:
33:
19:
4012:
4008:
3999:
3980:
3971:
3960:. Retrieved
3953:the original
3948:
3935:
3916:
3898:
3894:
3888:
3879:
3875:
3869:
3860:
3854:
3830:
3823:
3814:
3810:
3804:
3779:
3775:
3769:
3742:
3736:
3716:
3709:
3634:
3625:
3610:
3606:
3587:
3584:
3569:
3559:
3555:
3551:
3547:
3539:
3535:
3531:
3527:
3523:
3519:
3515:
3511:
3507:
3503:
3499:
3495:
3493:
3488:
3484:
3480:
3476:
3472:
3468:
3464:
3460:
3456:
3452:
3448:
3444:
3440:
3436:
3432:
3428:
3424:
3420:
3418:
3415:
3214:
3180:
3173:
3074:
2997:results in:
2936:
2932:
2930:
2853:
2793:
2759:
2722:
2463:
2151:
1845:
1839:
1746:
1395:
1268:
1093:
922:
919:
866:
860:
852:
747:
620:
609:
344:
160:
148:
139:gantry crane
129:
124:
118:
94:
85:
81:
77:
67:
65:
31:
29:
18:Nonholonomic
3592:solid angle
3530:located at
3502:returns to
984:-axis, and
157:Constraints
68:anholonomic
40:mathematics
4033:Categories
3962:2007-09-22
3895:Матем. Сб.
3702:References
3544:quaternion
878:See also:
86:anholonomy
78:integrable
48:parameters
3840:cite book
3834:. London.
3617:curvature
3376:ϕ
3307:θ
3270:
3261:θ
3232:
3223:θ
3191:π
3150:
3141:θ
3115:∈
3105:π
3091:π
3083:θ
3055:θ
3052:
3023:θ
3020:
3014:−
3011:θ
3008:
2959:θ
2956:
2947:−
2915:θ
2895:θ
2892:
2841:θ
2838:
2803:−
2774:θ
2771:
2740:θ
2737:
2686:ϕ
2683:∂
2679:∂
2674:−
2668:θ
2665:
2656:−
2647:θ
2644:∂
2640:∂
2616:θ
2613:
2604:−
2592:∂
2588:∂
2583:−
2568:ϕ
2565:∂
2561:∂
2528:θ
2525:∂
2521:∂
2516:−
2498:∂
2494:∂
2484:θ
2481:
2472:−
2449:ϕ
2438:γ
2412:θ
2401:β
2364:α
2338:θ
2335:
2326:−
2318:γ
2284:β
2250:α
2219:γ
2192:β
2165:α
2122:γ
2114:∂
2107:β
2099:∂
2093:−
2085:β
2077:∂
2070:γ
2062:∂
2049:α
2028:α
2020:∂
2013:γ
2005:∂
1999:−
1991:γ
1983:∂
1976:α
1968:∂
1955:β
1934:β
1926:∂
1919:α
1911:∂
1905:−
1897:α
1889:∂
1882:β
1874:∂
1861:γ
1758:∑
1725:ϕ
1713:θ
1665:θ
1662:
1653:−
1631:θ
1628:
1619:−
1571:˙
1568:ϕ
1555:˙
1552:θ
1539:˙
1523:˙
1498:θ
1495:
1486:−
1464:θ
1461:
1452:−
1366:θ
1363:
1354:˙
1351:ϕ
1342:−
1336:˙
1323:θ
1320:
1311:˙
1308:ϕ
1299:−
1293:˙
1207:θ
1204:
1195:˙
1192:ϕ
1179:θ
1176:
1167:˙
1164:ϕ
1133:˙
1117:˙
1064:ϕ
1059:θ
952:θ
932:ϕ
780:≥
767:−
722:…
670:δ
634:∑
477:…
362:∑
345:is a non-
313:…
233:…
221:∈
125:holonomic
92:in 1894.
3979:(1980).
3655:See also
3637:robotics
3631:Robotics
3582:forces.
3580:Coriolis
844:Examples
3784:Bibcode
3761:2206889
3621:torsion
867:exactly
145:History
121:plotter
62:Details
36:physics
3987:
3923:
3817:: 1–5.
3759:
3724:
3267:arctan
3229:arctan
3147:arctan
3133:(from
886:, and
698:
695:
692:
689:
453:
450:
447:
444:
134:turtle
106:metric
104:has a
3956:(PDF)
3945:(PDF)
3613:helix
3550:and −
3181:would
2206:, or
1846:first
42:is a
3985:ISBN
3921:ISBN
3846:link
3722:ISBN
3647:for
3643:and
3578:and
3558:and
3423:and
3406:here
1004:and
621:For
210:for
38:and
4017:doi
3815:XII
3792:doi
3747:doi
3635:In
3171:).
3049:tan
3017:cos
3005:sin
2953:cos
2937:all
2933:all
2889:sin
2854:can
2835:sin
2768:sin
2734:sin
2662:cos
2610:cos
2478:cos
2332:cos
1659:sin
1625:cos
1492:sin
1458:cos
1360:sin
1317:cos
1201:sin
1173:cos
34:in
4035::
3947:.
3907:^
3842:}}
3838:{{
3813:.
3790:.
3780:43
3778:.
3757:MR
3755:.
3651:.
3599:.
2179:,
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783:0.
618:.
353::
116:.
58:.
30:A
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4019::
3993:.
3965:.
3929:.
3899:4
3880:1
3863:.
3848:)
3798:.
3794::
3786::
3763:.
3749::
3730:.
3588:t
3560:R
3556:B
3552:q
3548:q
3546:(
3540:z
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3528:C
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3433:C
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3389:T
3382:]
3371:x
3365:[
3359:=
3355:u
3333:x
3330:=
3327:y
3287:)
3284:x
3280:/
3276:y
3273:(
3264:=
3241:)
3238:1
3235:(
3226:=
3199:4
3195:/
3159:)
3156:1
3153:(
3144:=
3119:Z
3112:n
3108:;
3102:n
3099:+
3094:4
3086:=
3061:1
3058:=
3029:0
3026:=
2985:r
2965:0
2962:=
2950:r
2867:r
2832:r
2812:0
2809:=
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2692:(
2671:)
2659:r
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2634:(
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2488:(
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2406:=
2397:u
2375:x
2372:d
2369:=
2360:u
2329:r
2323:=
2314:A
2292:0
2289:=
2280:A
2258:1
2255:=
2246:A
2215:A
2188:A
2161:A
2138:0
2135:=
2131:)
2118:u
2103:A
2081:u
2066:A
2055:(
2045:A
2041:+
2037:)
2024:u
2009:A
1987:u
1972:A
1961:(
1951:A
1947:+
1943:)
1930:u
1915:A
1893:u
1878:A
1867:(
1857:A
1826:2
1823:,
1820:1
1817:=
1814:r
1810:;
1807:0
1804:=
1799:s
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1791:d
1786:s
1783:r
1779:A
1773:n
1768:1
1765:=
1762:s
1731:)
1721:d
1709:d
1701:y
1697:d
1689:x
1685:d
1678:(
1671:)
1656:r
1648:0
1643:1
1638:0
1622:r
1614:0
1609:0
1604:1
1598:(
1593:=
1589:0
1585:=
1580:)
1536:y
1520:x
1511:(
1504:)
1489:r
1481:0
1476:1
1471:0
1455:r
1447:0
1442:0
1437:1
1431:(
1409:t
1405:d
1381:0
1377:=
1372:)
1345:r
1333:y
1302:r
1290:x
1281:(
1255:y
1235:x
1213:)
1186:r
1158:r
1152:(
1147:=
1142:)
1130:y
1114:x
1105:(
1077:T
1070:]
1054:y
1049:x
1043:[
1037:=
1033:u
1012:y
992:x
972:x
904:r
827:a
805:r
775:2
771:a
762:2
758:r
734:.
731:)
728:k
725:,
719:,
716:2
713:,
710:1
707:=
704:s
701:(
686:0
683:=
678:i
674:q
665:i
662:,
659:s
655:a
649:n
644:1
641:=
638:i
592:i
589:,
586:s
582:a
557:i
553:q
530:k
508:n
486:)
483:k
480:,
474:,
471:2
468:,
465:1
462:=
459:s
456:(
441:0
438:=
435:t
432:d
426:t
423:,
420:s
416:a
412:+
407:i
403:q
399:d
393:i
390:,
387:s
383:a
377:n
372:1
369:=
366:i
331:,
328:0
325:=
322:)
319:t
316:,
310:,
305:3
300:r
295:,
290:2
285:r
280:,
275:1
270:r
265:(
262:f
242:}
239:N
236:,
230:,
227:1
224:{
218:i
196:i
191:r
169:N
20:)
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