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Rigid body

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(Redirected fromRigid bodies)
Physical object which does not deform when forces or moments are exerted on it
The position of a rigid body is determined by the position of its center of mass and by itsattitude (at least six parameters in total).[1]
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Classical mechanics
F=dpdt{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}

Inphysics, arigid body, also known as arigid object,[2] is asolidbody in whichdeformation is zero or negligible, when a deforming pressure or deforming force is applied on it. Thedistance between any two givenpoints on a rigid body remains constant in time regardless of externalforces ormoments exerted on it. A rigid body is usually considered as acontinuous distribution ofmass.Mechanics of rigid bodies is a field within mechanics where motions and forces of objects are studied without considering effects that can cause deformation (as opposed tomechanics of materials, where deformable objects are considered).

In the study ofspecial relativity, a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near thespeed of light, where the mass is infinitely large. Inquantum mechanics, a rigid body is usually thought of as a collection ofpoint masses. For instance,molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (seeclassification of molecules as rigid rotors).

Principles

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Linear and angular position

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The position of a rigid body is theposition of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, provided that theirtime-invariant position relative to the three selected particles is known. However, typically a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by:

  1. thelinear position orposition of the body, namely the position of one of the particles of the body, specifically chosen as a reference point (typically coinciding with thecenter of mass orcentroid of the body), together with
  2. theangular position (also known asorientation, orattitude) of the body.

Thus, the position of a rigid body has two components:linear andangular, respectively.[3] The same is true for otherkinematic andkinetic quantities describing the motion of a rigid body, such as linear and angularvelocity,acceleration,momentum,impulse, andkinetic energy.[4]

The linearposition can be represented by avector with its tail at an arbitrary reference point inspace (the origin of a chosencoordinate system) and its tip at an arbitrary point of interest on the rigid body, typically coinciding with itscenter of mass orcentroid. This reference point may define the origin of acoordinate system fixed to the body.

There are several ways to numerically describe theorientation of a rigid body, including a set of threeEuler angles, aquaternion, or adirection cosine matrix (also referred to as arotation matrix). All these methods actually define the orientation of abasis set (orcoordinate system) which has a fixed orientation relative to the body (i.e. rotates together with the body), relative to another basis set (or coordinate system), from which the motion of the rigid body is observed. For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of three orthogonalunit vectorsb1,b2,b3, such thatb1 is parallel to the chord line of the wing and directed forward,b2 is normal to the plane of symmetry and directed rightward, andb3 is given by the cross productb3=b1×b2{\displaystyle b_{3}=b_{1}\times b_{2}}.

In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to astranslation androtation, respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation (roto-translation) of the body starting from a hypothetic reference position (not necessarily coinciding with a position actually taken by the body during its motion).

Linear and angular velocity

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Velocity (also calledlinear velocity) andangular velocity are measured with respect to aframe of reference.

The linear velocity of a rigid body is avector quantity, equal to thetime rate of change of its linear position. Thus, it is the velocity of a reference point fixed to the body. During purely translational motion (motion with no rotation), all points on a rigid body move with the samevelocity. However, whenmotion involves rotation, the instantaneous velocity of any two points on the body will generally not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneousaxis of rotation.

Angular velocity is a vector quantity that describes theangular speed at which the orientation of the rigid body is changing and the instantaneousaxis about which it is rotating (the existence of this instantaneous axis is guaranteed by theEuler's rotation theorem). All points on a rigid body experience the sameangular velocity at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneousaxis of rotation. The relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not thetime rate of change of orientation, because there is no such concept as an orientation vector that can bedifferentiated to obtain the angular velocity.

Kinematical equations

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Addition theorem for angular velocity

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Theangular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect to D:[5]

NωB=NωD+DωB.{\displaystyle {}^{\mathrm {N} }\!{\boldsymbol {\omega }}^{\mathrm {B} }={}^{\mathrm {N} }\!{\boldsymbol {\omega }}^{\mathrm {D} }+{}^{\mathrm {D} }\!{\boldsymbol {\omega }}^{\mathrm {B} }.}

In this case, rigid bodies and reference frames are indistinguishable and completely interchangeable.

Addition theorem for position

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For any set of three points P, Q, and R, the position vector from P to R is the sum of the position vector from P to Q and the position vector from Q to R:

rPR=rPQ+rQR.{\displaystyle \mathbf {r} ^{\mathrm {PR} }=\mathbf {r} ^{\mathrm {PQ} }+\mathbf {r} ^{\mathrm {QR} }.}

The norm of a position vector is the spatial distance.Here the coordinates of all three vectors must be expressed in coordinate frames with the same orientation.

Mathematical definition of velocity

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The velocity of point P in reference frame N is defined as thetime derivative in N of the position vector from O to P:[6]

NvP=Nddt(rOP){\displaystyle {}^{\mathrm {N} }\mathbf {v} ^{\mathrm {P} }={\frac {{}^{\mathrm {N} }\mathrm {d} }{\mathrm {d} t}}(\mathbf {r} ^{\mathrm {OP} })}

where O is any arbitrary point fixed in reference frame N, and the N to the left of the d/dt operator indicates that the derivative is taken in reference frame N. The result is independent of the selection of O so long as O is fixed in N.

Mathematical definition of acceleration

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The acceleration of point P in reference frame N is defined as thetime derivative in N of its velocity:[6]

NaP=Nddt(NvP).{\displaystyle {}^{\mathrm {N} }\mathbf {a} ^{\mathrm {P} }={\frac {^{\mathrm {N} }\mathrm {d} }{\mathrm {d} t}}({}^{\mathrm {N} }\mathbf {v} ^{\mathrm {P} }).}

Velocity of two points fixed on a rigid body

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For two points P and Q that are fixed on a rigid body B, where B has an angular velocityNωB{\displaystyle \scriptstyle {^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }}} in the reference frame N, the velocity of Q in N can be expressed as a function of the velocity of P in N:[7]

NvQ=NvP+NωB×rPQ.{\displaystyle {}^{\mathrm {N} }\mathbf {v} ^{\mathrm {Q} }={}^{\mathrm {N} }\!\mathbf {v} ^{\mathrm {P} }+{}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }\times \mathbf {r} ^{\mathrm {PQ} }.}

whererPQ{\displaystyle \mathbf {r} ^{\mathrm {PQ} }} is the position vector from P to Q.,[7] with coordinates expressed in N (or a frame with the same orientation as N.) This relation can be derived from the temporal invariance of the norm distance between P and Q.

Acceleration of two points fixed on a rigid body

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Bydifferentiating the equation for theVelocity of two points fixed on a rigid body in N with respect to time, the acceleration in reference frame N of a point Q fixed on a rigid body B can be expressed as

NaQ=NaP+NωB×(NωB×rPQ)+NαB×rPQ{\displaystyle {}^{\mathrm {N} }\mathbf {a} ^{\mathrm {Q} }={}^{\mathrm {N} }\mathbf {a} ^{\mathrm {P} }+{}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }\times \left({}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }\times \mathbf {r} ^{\mathrm {PQ} }\right)+{}^{\mathrm {N} }{\boldsymbol {\alpha }}^{\mathrm {B} }\times \mathbf {r} ^{\mathrm {PQ} }}

whereNαB{\displaystyle \scriptstyle {{}^{\mathrm {N} }\!{\boldsymbol {\alpha }}^{\mathrm {B} }}} is theangular acceleration of B in the reference frame N.[7]

Angular velocity and acceleration of two points fixed on a rigid body

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As mentionedabove, all points on a rigid body B have the same angular velocityNωB{\displaystyle {}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }} in a fixed reference frame N, and thus the same angular accelerationNαB.{\displaystyle {}^{\mathrm {N} }{\boldsymbol {\alpha }}^{\mathrm {B} }.}

Velocity of one point moving on a rigid body

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If the point R is moving in the rigid body B while B moves in reference frame N, then the velocity of R in N is

NvR=NvQ+BvR{\displaystyle {}^{\mathrm {N} }\mathbf {v} ^{\mathrm {R} }={}^{\mathrm {N} }\mathbf {v} ^{\mathrm {Q} }+{}^{\mathrm {B} }\mathbf {v} ^{\mathrm {R} }}

where Q is the point fixed in B that is instantaneously coincident with R at the instant of interest.[8] This relation is often combined with the relation for theVelocity of two points fixed on a rigid body.

Acceleration of one point moving on a rigid body

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The acceleration in reference frame N of the point R moving in body B while B is moving in frame N is given by

NaR=NaQ+BaR+2NωB×BvR{\displaystyle {}^{\mathrm {N} }\mathbf {a} ^{\mathrm {R} }={}^{\mathrm {N} }\mathbf {a} ^{\mathrm {Q} }+{}^{\mathrm {B} }\mathbf {a} ^{\mathrm {R} }+2{}^{\mathrm {N} }{\boldsymbol {\omega }}^{\mathrm {B} }\times {}^{\mathrm {B} }\mathbf {v} ^{\mathrm {R} }}

where Q is the point fixed in B that instantaneously coincident with R at the instant of interest.[8] This equation is often combined withAcceleration of two points fixed on a rigid body.

Other quantities

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IfC is the origin of a localcoordinate systemL, attached to the body,thespatial ortwistacceleration of a rigid body is defined as thespatial acceleration ofC (as opposed to material acceleration above):ψ(t,r0)=a(t,r0)−ω(t)×v(t,r0)=ψc(t)+α(t)×A(t)r0{\displaystyle {\boldsymbol {\psi }}(t,\mathbf {r} _{0})=\mathbf {a} (t,\mathbf {r} _{0})-{\boldsymbol {\omega }}(t)\times \mathbf {v} (t,\mathbf {r} _{0})={\boldsymbol {\psi }}_{c}(t)+{\boldsymbol {\alpha }}(t)\times A(t)\mathbf {r} _{0}}where

In 2D, the angular velocity is a scalar, and matrix A(t) simply represents a rotation in thexy-plane by an angle which is the integral of the angular velocity over time.

Vehicles, walking people, etc., usually rotate according to changes in the direction of the velocity: they move forward with respect to their own orientation. Then, if the body follows a closed orbit in a plane, the angular velocity integrated over a time interval in which the orbit is completed once, is an integer times 360°. This integer is thewinding number with respect to the origin of the velocity. Compare theamount of rotation associated with the vertices of a polygon.

Instantaneous rotation axis formulae

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Assume thatv(P){\displaystyle \mathbf {v} (\mathbf {P} )} is a smooth 3-d vector field andO{\displaystyle O} is a point inR3{\displaystyle \mathbb {R} ^{3}},withvO=v(O){\displaystyle \mathbf {v} _{O}=\mathbf {v} (O)}. DenoteBε{\displaystyle B_{\varepsilon }} the ball of radiusε{\displaystyle \varepsilon } centered atO{\displaystyle O}, andr=P−O{\displaystyle \mathbf {r} =\mathbf {P} -O}. We examine the expressionIε=∫Bϵr×(v(P)−vO)r2dV.{\displaystyle \mathbf {I} _{\varepsilon }=\int _{B_{\epsilon }}{\frac {\mathbf {r} \times (\mathbf {v} (\mathbf {P} )-\mathbf {v} _{O})}{r^{2}}}\,dV.}

Linearizing the velocity field atO{\displaystyle O} givesv(P)−vO=(∇v)Or+o(r),{\displaystyle \mathbf {v} (\mathbf {P} )-\mathbf {v} _{O}=(\nabla \mathbf {v} )_{O}\,\mathbf {r} +o(r),}where(∇v)O{\displaystyle (\nabla \mathbf {v} )_{O}} is the Jacobian matrix atO{\displaystyle O}.

Decompose it into symmetric and antisymmetric parts:(∇v)O=Js+Ja{\displaystyle (\nabla \mathbf {v} )_{O}=J_{s}+J_{a}}, withJa{\displaystyle J_{a}} antisymmetric. By linear algebra, there exists a vectorw{\displaystyle {\boldsymbol {w}}} such thatJar=w×r{\displaystyle J_{a}\mathbf {r} ={\boldsymbol {w}}\times \mathbf {r} }. In fact, direct computation shows thatw=12∇×v(O){\displaystyle {\boldsymbol {w}}={1 \over 2}\nabla \times \mathbf {v} (O)}.The symmetric partJs{\displaystyle J_{s}} does not contribute to the integral, hence

Iε=∫Bεr×(Jar)r2dV+o(ε3)=∫Bεr×(w×r)r2dV+o(ε3).{\displaystyle \mathbf {I} _{\varepsilon }=\int _{B_{\varepsilon }}{\frac {\mathbf {r} \times (J_{a}\mathbf {r} )}{r^{2}}}\,dV+o(\varepsilon ^{3})=\int _{B_{\varepsilon }}{\frac {\mathbf {r} \times ({\boldsymbol {w}}\times \mathbf {r} )}{r^{2}}}\,dV+o(\varepsilon ^{3}).}

Using the triple product identity, there holdsr×(w×r)r2=w−(r⋅w)rr2.{\displaystyle {\frac {\mathbf {r} \times ({\boldsymbol {w}}\times \mathbf {r} )}{r^{2}}}={\boldsymbol {w}}-{\frac {(\mathbf {r} \cdot {\boldsymbol {w}})\mathbf {r} }{r^{2}}}.}

Integrating over the ball and using spherical symmetry,∫Bε(r⋅w)rr2dV=13Vol(Bε)w,{\displaystyle \int _{B_{\varepsilon }}{\frac {(\mathbf {r} \cdot {\boldsymbol {w}})\mathbf {r} }{r^{2}}}\,dV={\frac {1}{3}}{\text{Vol}}(B_{\varepsilon }){\boldsymbol {w}},}so thatIε=23Vol(Bε)w+o(ε3),withw=12∇×v(O).(∗){\displaystyle \mathbf {I} _{\varepsilon }={\frac {2}{3}}{\text{Vol}}(B_{\varepsilon }){\boldsymbol {w}}+o(\varepsilon ^{3}),\quad {\rm {with}}\quad {\boldsymbol {w}}={1 \over 2}\nabla \times \mathbf {v} (O).\quad (*)}

Incidentally, this formula provides an integral formulation of the curl of the vector field atO{\displaystyle O}:∇×v(O)=limε→03Vol(Bε)∫Bϵr×(v(P)−vO)r2dV.{\displaystyle \nabla \times \mathbf {v} (O)=\lim _{\varepsilon \to 0}{3 \over {\text{Vol}}(B_{\varepsilon })}\int _{B_{\epsilon }}{\frac {\mathbf {r} \times (\mathbf {v} (\mathbf {P} )-\mathbf {v} _{O})}{r^{2}}}\,dV.}

Coordinate free formula for the instantaneous rotation vector

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Now, assume a rigid body is rotating with angular velocityω{\displaystyle {\boldsymbol {\omega }}}.By rigid body kinematics, using the notations above, the field of velocities is given at every timet{\displaystyle t} byv(P)=vO+ω×r.{\displaystyle \mathbf {v} (\mathbf {P} )=\mathbf {v} _{O}+{\boldsymbol {\omega }}\times \mathbf {r} .}Thus, the vector fieldv(P)−vO{\displaystyle \mathbf {v} (\mathbf {\mathbf {P} } )-\mathbf {v} _{O}} is linear inr{\displaystyle \mathbf {r} }. It follows that(∇v)Or=ω×r=Ja{\displaystyle (\nabla \mathbf {v} )_{O}\,\mathbf {r} ={\boldsymbol {\omega }}\times \mathbf {r} =J_{a}}.Thusω=w{\displaystyle {\boldsymbol {\omega }}={\boldsymbol {w}}} and the termso(r){\displaystyle o(r)} ando(ε3){\displaystyle o(\varepsilon ^{3})} vanish identically in the above formulae.Therefore(∗){\displaystyle (*)} impliesIε=23Vol(Bε)ω.{\displaystyle \mathbf {I} _{\varepsilon }={\frac {2}{3}}\mathrm {Vol} (B_{\varepsilon })\,{\boldsymbol {\omega }}.}Solving forω{\displaystyle {\boldsymbol {\omega }}} yields, for every ballBε{\displaystyle B_{\varepsilon }} centered atO{\displaystyle O},ω=32Vol(Bε)∫Bεr×(v(P)−vO)r2dV.{\displaystyle {\boldsymbol {\omega }}={\frac {3}{2\,\mathrm {Vol} (B_{\varepsilon })}}\int _{B_{\varepsilon }}{\frac {\mathbf {r} \times (\mathbf {v} (\mathbf {P} )-\mathbf {v} _{O})}{r^{2}}}\,dV.}

Curl formula

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From(∗){\displaystyle (*)} and the fact thato(ε3){\displaystyle o(\varepsilon ^{3})} vanishes identically (as seen just above), the curl formula follows:ω=12∇×v(O).{\displaystyle {\boldsymbol {\omega }}={\frac {1}{2}}\nabla \times \mathbf {v} (O).}

Kinetics

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Main article:Rigid body dynamics

Any point that is rigidly connected to the body can be used as reference point (origin of coordinate systemL) to describe the linear motion of the body (the linear position, velocity and acceleration vectors depend on the choice).

However, depending on the application, a convenient choice may be:

  • thecenter of mass of the whole system, which generally has the simplest motion for a body moving freely in space;
  • a point such that the translational motion is zero or simplified, e.g. on anaxle orhinge, at the center of aball and socket joint, etc.

When the center of mass is used as reference point:

  • The (linear)momentum is independent of the rotational motion. At any time it is equal to the total mass of the rigid body times the translational velocity.
  • Theangular momentum with respect to the center of mass is the same as without translation: at any time it is equal to theinertia tensor times the angular velocity. When the angular velocity is expressed with respect to a coordinate system coinciding with theprincipal axes of the body, each component of the angular momentum is a product of a moment of inertia (a principal value of the inertia tensor) times the corresponding component of the angular velocity; thetorque is the inertia tensor times theangular acceleration.
  • Possible motions in the absence of external forces are translation with constant velocity, steady rotation about a fixed principal axis, and also torque-freeprecession.
  • The net external force on the rigid body is always equal to the total mass times the translational acceleration (i.e.,Newton's second law holds for the translational motion, even when the net external torque is nonzero, and/or the body rotates).
  • The totalkinetic energy is simply the sum of translational androtational energy.

Geometry

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Two rigid bodies are said to bedifferent (not copies) if there is noproper rotation from one to the other. A rigid body is calledchiral if itsmirror image is different in that sense, i.e., if it has either nosymmetry or itssymmetry group contains only proper rotations. In the opposite case an object is called achiral: the mirror image is a copy, not a different object. Such an object may have a symmetry plane, but not necessarily: there may also be a plane of reflection with respect to which the image of the object is a rotated version. The latter applies forS2n, of which the casen = 1 is inversion symmetry.

For a (rigid) rectangular transparent sheet, inversion symmetry corresponds to having on one side an image without rotational symmetry and on the other side an image such that what shines through is the image at the top side, upside down. We can distinguish two cases:

  • the sheet surface with the image is not symmetric - in this case the two sides are different, but the mirror image of the object is the same, after a rotation by 180° about the axis perpendicular to the mirror plane.
  • the sheet surface with the image has a symmetry axis - in this case the two sides are the same, and the mirror image of the object is also the same, again after a rotation by 180° about the axis perpendicular to the mirror plane.

A sheet with a through and through image is achiral. We can distinguish again two cases:

  • the sheet surface with the image has no symmetry axis - the two sides are different
  • the sheet surface with the image has a symmetry axis - the two sides are the same

Configuration space

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Theconfiguration space of a rigid body with one point fixed (i.e., a body with zero translational motion) is given by the underlyingmanifold of therotation group SO(3). The configuration space of a nonfixed (with non-zero translational motion) rigid body isE+(3), the subgroup ofdirect isometries of theEuclidean group in three dimensions (combinations oftranslations androtations).

See also

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Notes

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  1. ^Lorenzo Sciavicco, Bruno Siciliano (2000)."§2.4.2 Roll-pitch-yaw angles".Modelling and control of robot manipulators (2nd ed.). Springer. p. 32.ISBN 1-85233-221-2.
  2. ^Andy Ruina and Rudra Pratap (2015).Introduction to Statics and Dynamics. Oxford University Press. (link:[1])
  3. ^In general, the position of a point or particle is also known, in physics, aslinear position, as opposed to theangular position of a line, or line segment (e.g., incircular motion, the "radius" joining the rotating point with the center of rotation), orbasis set, orcoordinate system.
  4. ^Inkinematics,linear means "along a straight or curved line" (the path of the particle inspace). Inmathematics, however,linear has a different meaning. In both contexts, the word "linear" is related to the word "line". In mathematics, aline is often defined as a straightcurve. For those who adopt this definition, acurve can be straight, and curved lines are not supposed to exist. Inkinematics, the termline is used as a synonym of the termtrajectory, orpath (namely, it has the same non-restricted meaning as that given, in mathematics, to the wordcurve). In short, both straight and curved lines are supposed to exist. In kinematics anddynamics, the following words refer to the same non-restricted meaning of the term "line":
    • "linear" (= along a straight or curved line),
    • "rectilinear" (= along a straight line, from Latinrectus = straight, andlinere = spread),
    • "curvilinear" (=along a curved line, from Latincurvus = curved, andlinere = spread).
    Intopology andmeteorology, the term "line" has the same meaning; namely, acontour line is a curve.
  5. ^Kane, Thomas; Levinson, David (1996). "2-4 Auxiliary Reference Frames".Dynamics Online. Sunnyvale, California: OnLine Dynamics, Inc.
  6. ^abKane, Thomas; Levinson, David (1996). "2-6 Velocity and Acceleration".Dynamics Online. Sunnyvale, California: OnLine Dynamics, Inc.
  7. ^abcKane, Thomas; Levinson, David (1996). "2-7 Two Points Fixed on a Rigid Body".Dynamics Online. Sunnyvale, California: OnLine Dynamics, Inc.
  8. ^abKane, Thomas; Levinson, David (1996). "2-8 One Point Moving on a Rigid Body".Dynamics Online. Sunnyvale, California: OnLine Dynamics, Inc.

References

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  • Roy Featherstone (1987).Robot Dynamics Algorithms. Springer.ISBN 0-89838-230-0. This reference effectively combinesscrew theory with rigid bodydynamics for robotic applications. The author also chooses to usespatial accelerations extensively in place of material accelerations as they simplify the equations and allow for compact notation.
  • JPL DARTS page has a section on spatial operator algebra (link:[2]) as well as an extensive list of references (link:[3]).
  • Andy Ruina and Rudra Pratap (2015).Introduction to Statics and Dynamics. Oxford University Press. (link:[4]).
  • Prof. Dr. Dennis M. Kochmann, Dynamics Lecture Notes, ETH Zurich.[5]

External links

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