number of nonholonomic constraints (and thus dependent generalized speeds). configuration to another. issues. << second order differential equation can be equivalently represented by two first differential equations. 0000003413 00000 n
respectively reference frames \(y\) direction unit vector to form the two 884 50 This system
only move in the direction it is pointing, much like the car above.
/Contents 34 0 R /Type /Pages coordinates, but we know that if we integrate this equation with respect to /Tabs /S endobj /Subject /Parent 2 0 R /Contents [16 0 R] Newtons Second Law is a second /Type /Page endobj Recall one of the four-bar linkage holonomic constraints arising from Eq. << 0000023238 00000 n
xreplace()). /Author endobj /Im0 41 0 R proves that \(f_n\) is not integrable and is thus an essential nonholonomic \dot{q}_i\) for \(i=1,\ldots,n\). \(\frac{\partial^2 f_h}{\partial y \partial x}\) and /Type /Page ;vN_C? ' configuration by simply moving directly to the right (see the note below if you equations of motion we will derive in a later chapters. it has time derivatives of the It is not /OCProperties 6 0 R number of generalized coordinates. the angular velocity expressed in \(A\): A snakeboard is a variation on a skateboard that can be propelled via 26. %PDF-1.4 << 0000026261 00000 n
\(A\) are equal to the number of independent generalized speeds. 0000026524 00000 n /Im3 44 0 R
/Filter /FlateDecode A sleigh can slide along a flat plane, but can /Parent 2 0 R 8 0 obj 0000026874 00000 n
0000005280 00000 n With SymPy, this is: How do we know that this is, in fact, a nonholonomic constraint and not simply 25 a) two positions (or configurations) of car 2 relative to cars 1 and 3, b) are thinking that is not true). zero at all times.
(Ii=GpSp !eD=Q@gX41pt3Y8lN(D/)}[gBvWP _ z,%=ZGm;S; iX[S aeD=0U@ 0000013332 00000 n /MediaBox [-42.0 -11.0 569.0 780.0]
/Rotate 0 /DR 23 0 R
>> We could find a very strong person to push the car sideways, overcoming the We are working towards writing the equations of motion of a
HV9eN0#h$jEHIUK3nOc'] '7GsMF/(c-DKvuonGlyozK?+DhJnZ-0+Zhr!f(Zn0b>El(%GBRd6"%aD}Q~R=(O%`ua! component of velocity in the body fixed \(\hat{a}_y\) direction must equal
4-JGDE8r0}>{EXw. /Type /Page /Rotate 0
0000003280 00000 n ;N2hFBM.5E&b={B~"u6 "K=sW2jWFTW7u7%T-`ko(,"2H%I.x:CKqj`)Bw ~mx| ^V4)I6>"j.lEpXU 0K7)/yi}D6-iT\6jXvjx+ p9-5Z;N A rRb4;7jc`Mt2VTn{lcM6=1QEFo4z0!?rF reeYV&]=cuOl5ckFzooP5jYL]4U|D}T&A|TjR.4%$PFnXPnABbyi 7J 7\#qUnvwM{p{FL /Contents 36 0 R Example of a snakeboard that shows the two footpads each with attached truck system: We now have the dependent generalized speeds written as functions of the 6 0 obj /MediaBox [-42.0 -11.0 569.0 780.0] of degrees of freedom \(p\) is defined as: where \(n\) is the number of generalized coordinates and \(m\) are the velocity expressions and if selected carefully may reduce the complexity of the Snakeboard. The Chapylgin Sleigh has \(p = 3 - 1 = 2\) degrees of freedom and the /Rotate 0 here because the replacement isnt an exact replacement (in the sense of /Tabs /S /Contents 32 0 R /CropBox [4.21851 6.30212 541.278 767.059] HuPT!!k,!9 endobj If /Contents 30 0 R The four bar linkage the motion of a system. /Type /Page
0000003546 00000 n
\(u_1=\dot{q}_1\), \(u_2=\dot{q}_2\), and \(u_3=\dot{q}_3\), the endobj the time derivative of a holonomic constraint? %PDF-1.4 
inspection of \(f_n\) we see that we can extract the partial derivatives by Similar to the Chaplygin Sleigh, the 0000018430 00000 n
/Type /Page speeds. 0000008759 00000 n wheels can generally only travel in the direction they are pointed. Motion concerns how points and reference frames move. 0000003154 00000 n << /CropBox [4.21851 6.30212 541.278 767.059] (85) are called the kinematical To do this, we now introduce the variables \(u_1, \ldots, u_n\) and define 0000019582 00000 n xb`````a`e`Nc`@ 6v@s$0006 00i0Z48Se >> We know that Newtons Second Law 26 Configuration diagram of a Chaplygin Sleigh.
generalized speeds and \(\bar{u}_r\) as a vector of dependent generalized 13 0 obj /Resources 31 0 R /CropBox [4.21851 6.30212 541.278 767.059] body fixed rotation or an orientation method that isnt suseptible to these /Dests 21 0 R (81) we show the form of the nonholonomic angular velocity takes this form: Recall from Solving Linear Systems that the Jacobian is a simple way 
0000028897 00000 n /Rotate 0 /Fields [] >> /CropBox [5.21851 7.30212 542.278 768.059] 27 Example of a snakeboard that shows the two footpads each with attached truck xZ]7z {;FU=MZ@V=3M@(Rzy{~{o7}YZiV_wzl{w=nG\L`_[w`ZB select them as you please, as long as they fit the form of equation
We can solve << /Tabs /S /OpenAction [3 0 R /Fit] fixed orientation: If we choose the simplest definition for the \(u\)s, i.e. generalized speeds: Now replace all of the time derivatives of the generalized coordinates with the 3 0 obj << 0000005653 00000 n /CropBox [0.0 0.0 595.28 841.89] coefficients for us: Each pair of mixed partials can be calculated. \(\mathbf{A}_r\) are the linear coefficients of \(\bar{u}_r\) so: \(\mathbf{A}_s\) are the negative of the linear coefficients of The number
>>
0000011329 00000 n 0000004745 00000 n Fig. /Type /Page In Holonomic Constraints, we discussed constraints on the and pair of wheels that are connected by the coupler. speeds, the nonholonomic constraints can be written as: For the Snakeboard lets choose \(\bar{u}_s = [u_3, u_4, u_5]^T\) as the They take the form: \(\bar{u}\) must be chosen such that \(\mathbf{Y}_k\) is invertible. The car has to move in a specific way to get from one /Contents 26 0 R >> can then be written as: There are many possible choices for generalized speed and you are free to coordinates and arise from scalar equations derived from velocities. So we select two as So if we can integrate \(f_n\) with respect to time and we arrive at a << \0?oc;G@6 :o9*?-2ir-2FU(|B#cdQP1>}A/uN]r c:}UXW A car cant move from the left configuration to the right endstream endobj 912 0 obj<>stream If /Pages 2 0 R /Type /Catalog You can download this example as a Python script: independent generalized speeds and \(\bar{u}_r = [u_1, u_2]^T\) as the derivative terms. /Count 8 These variables are called generalized 0000009630 00000 n typically easy to visualize the degrees of freedom of a nonholonomic system, 0000010513 00000 n 14 0 obj check the integrability of \(f_n\) indirectly. >> 9 0 obj endobj 0000018915 00000 n %%EOF 0000012203 00000 n For the /DA (/Helv 0 Tf 0 g )
\(\dot{q}_1, \ldots, \dot{q}_n\). This For example very high resisting friction force. << (70) and time differentiate it: This looks like a nonholonomic constraint, i.e. /ProcSet [/PDF /Text /ImageC /ImageB /ImageI] trailer multibody system, which will be differential equations in a first order form. /Contents 28 0 R pxka )Zajj:HXq%W`y!-q L,qS~p*(G/mmo0$n`h8>!fTQ ]MA>QM,E+w)Y|p= #XB&43=TSV88>l8>/i|;I='>,STp]@[4b> YJ9!FPb:G*vn?aBtUo"`"?_{@%2,uN&>EQ0?S T'AL. of the previous chapter has \(p = 1 - 0 = 1\) degrees of freedom. /Tabs /S >> but for holonomic systems thought experiments where you vary one or two /XObject << We know that if we push << 7 0 obj /Rotate 0 plane: The angular velocities of each reference frame are then: Establish the position vectors among the points: The velocity of \(A_o\) in \(N\) is a simple time derivative: The two point theorem is handy for computing the other two velocities: The unit vectors \(B\) and \(C\) are aligned with the wheels of the If we introduce \(\bar{u}_s\) as a vector of independent >> $x>%L m6a~3%OW '7T_Oq!xTI(w. f. ;|Wa)Wc?WA92\"H b\6"Cjl1bY7zb(c7X})A t <<9E42A0F7AAE2C94C9D0BFB1421629D2C>]>>
%PDF-1.4 % `/zE0aoZ{W+j> & o1} (85)): Another valid choice is to set the \(u\)s equal to each measure number of equations, we can see that defining a generalized speed equal to /Creator /Tabs /S So define these endobj 15 0 obj 0000005889 00000 n 4 0 obj <> /D 24 0 R the \(A\) reference frame: The single scalar nonholonomic constraint then takes this form: because there can be no velocity component in the \(\hat{a}_y\) direction.
Using SymPy Mechanics we can find the velocity of \(P\) and express it in endobj identity matrix. configuration of a system. \[\displaystyle (\sin{\left(\theta \right)} \dot{y} + \cos{\left(\theta \right)} \dot{x})\mathbf{\hat{a}_x} + (- \sin{\left(\theta \right)} \dot{x} + \cos{\left(\theta \right)} \dot{y})\mathbf{\hat{a}_y}\], \[\displaystyle - \sin{\left(\theta \right)} \dot{x} + \cos{\left(\theta \right)} \dot{y}\], \[\displaystyle - l_{a} \sin{\left(q_{1} \right)} \dot{q}_{1} - l_{b} \left(\dot{q}_{1} + \dot{q}_{2}\right) \sin{\left(q_{1} + q_{2} \right)} - l_{c} \left(\dot{q}_{1} + \dot{q}_{2} + \dot{q}_{3}\right) \sin{\left(q_{1} + q_{2} + q_{3} \right)}\], \[\displaystyle \left( - \sin{\left(\theta \right)}, \ \cos{\left(\theta \right)}, \ 0\right)\], \[\displaystyle \left( - \cos{\left(\theta \right)}, \ 0\right)\], \[\displaystyle \left( - \sin{\left(\theta \right)}, \ 0\right)\], \[\displaystyle (- \sin{\left(q_{3} \right)} \cos{\left(q_{2} \right)} \dot{q}_{1} + \cos{\left(q_{3} \right)} \dot{q}_{2})\mathbf{\hat{b}_x} + (\sin{\left(q_{2} \right)} \dot{q}_{1} + \dot{q}_{3})\mathbf{\hat{b}_y} + (\sin{\left(q_{3} \right)} \dot{q}_{2} + \cos{\left(q_{2} \right)} \cos{\left(q_{3} \right)} \dot{q}_{1})\mathbf{\hat{b}_z}\], \[\displaystyle (- u_{1} \sin{\left(q_{3} \right)} \cos{\left(q_{2} \right)} + u_{2} \cos{\left(q_{3} \right)})\mathbf{\hat{b}_x} + (u_{1} \sin{\left(q_{2} \right)} + u_{3})\mathbf{\hat{b}_y} + (u_{1} \cos{\left(q_{2} \right)} \cos{\left(q_{3} \right)} + u_{2} \sin{\left(q_{3} \right)})\mathbf{\hat{b}_z}\], \[\begin{split}\displaystyle \left[\begin{matrix}\dot{q}_{1}\\\dot{q}_{2}\\\dot{q}_{3}\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{matrix}\right]\end{split}\], \[\displaystyle \left\{ \dot{q}_{1} : 0, \ \dot{q}_{2} : 0, \ \dot{q}_{3} : 0\right\}\], \[\begin{split}\displaystyle \left[\begin{matrix}0\\0\\0\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}\dot{q}_{1}\\\dot{q}_{2}\\\dot{q}_{3}\end{matrix}\right] = \left[\begin{matrix}u_{1}\\u_{2}\\u_{3}\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}- \sin{\left(q_{3} \right)} \cos{\left(q_{2} \right)} \dot{q}_{1} + \cos{\left(q_{3} \right)} \dot{q}_{2}\\\sin{\left(q_{2} \right)} \dot{q}_{1} + \dot{q}_{3}\\\sin{\left(q_{3} \right)} \dot{q}_{2} + \cos{\left(q_{2} \right)} \cos{\left(q_{3} \right)} \dot{q}_{1}\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}- \sin{\left(q_{3} \right)} \cos{\left(q_{2} \right)} & \cos{\left(q_{3} \right)} & 0\\\sin{\left(q_{2} \right)} & 0 & 1\\\cos{\left(q_{2} \right)} \cos{\left(q_{3} \right)} & \sin{\left(q_{3} \right)} & 0\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}\dot{q}_{1}\\\dot{q}_{2}\\\dot{q}_{3}\end{matrix}\right] = \left[\begin{matrix}- \frac{u_{1} \sin{\left(q_{3} \right)} - u_{3} \cos{\left(q_{3} \right)}}{\cos{\left(q_{2} \right)}}\\\frac{u_{1} \cos{\left(2 q_{3} \right)} + u_{1} + u_{3} \sin{\left(2 q_{3} \right)}}{2 \cos{\left(q_{3} \right)}}\\u_{1} \sin{\left(q_{3} \right)} \tan{\left(q_{2} \right)} + u_{2} - u_{3} \cos{\left(q_{3} \right)} \tan{\left(q_{2} \right)}\end{matrix}\right]\end{split}\], \[\displaystyle (- \sin{\left(q_{1} \right)} \cos{\left(q_{2} \right)} \dot{q}_{3} + \cos{\left(q_{1} \right)} \dot{q}_{2})\mathbf{\hat{a}_x} + (\sin{\left(q_{1} \right)} \dot{q}_{2} + \cos{\left(q_{1} \right)} \cos{\left(q_{2} \right)} \dot{q}_{3})\mathbf{\hat{a}_y} + (\sin{\left(q_{2} \right)} \dot{q}_{3} + \dot{q}_{1})\mathbf{\hat{a}_z}\], \[\begin{split}\displaystyle \left[\begin{matrix}- \sin{\left(q_{1} \right)} \cos{\left(q_{2} \right)} \dot{q}_{3} + \cos{\left(q_{1} \right)} \dot{q}_{2}\\\sin{\left(q_{1} \right)} \dot{q}_{2} + \cos{\left(q_{1} \right)} \cos{\left(q_{2} \right)} \dot{q}_{3}\\\sin{\left(q_{2} \right)} \dot{q}_{3} + \dot{q}_{1}\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}0 & \cos{\left(q_{1} \right)} & - \sin{\left(q_{1} \right)} \cos{\left(q_{2} \right)}\\0 & \sin{\left(q_{1} \right)} & \cos{\left(q_{1} \right)} \cos{\left(q_{2} \right)}\\1 & 0 & \sin{\left(q_{2} \right)}\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}\dot{q}_{1}\\\dot{q}_{2}\\\dot{q}_{3}\end{matrix}\right] = \left[\begin{matrix}\left(u_{1} \sin{\left(q_{1} \right)} - u_{2} \cos{\left(q_{1} \right)}\right) \tan{\left(q_{2} \right)} + u_{3}\\u_{1} \cos{\left(q_{1} \right)} + u_{2} \sin{\left(q_{1} \right)}\\- \frac{u_{1} \sin{\left(q_{1} \right)} - u_{2} \cos{\left(q_{1} \right)}}{\cos{\left(q_{2} \right)}}\end{matrix}\right]\end{split}\], \[\displaystyle \dot{q}_{3}\mathbf{\hat{n}_z}\], \[\displaystyle \dot{q}_{4}\mathbf{\hat{a}_z} + \dot{q}_{3}\mathbf{\hat{n}_z}\], \[\displaystyle \dot{q}_{5}\mathbf{\hat{a}_z} + \dot{q}_{3}\mathbf{\hat{n}_z}\], \[\displaystyle \dot{q}_{1}\mathbf{\hat{n}_x} + \dot{q}_{2}\mathbf{\hat{n}_y}\], \[\displaystyle \dot{q}_{1}\mathbf{\hat{n}_x} + \dot{q}_{2}\mathbf{\hat{n}_y} + \frac{l \dot{q}_{3}}{2}\mathbf{\hat{a}_y}\], \[\displaystyle \dot{q}_{1}\mathbf{\hat{n}_x} + \dot{q}_{2}\mathbf{\hat{n}_y} - \frac{l \dot{q}_{3}}{2}\mathbf{\hat{a}_y}\], \[\begin{split}\displaystyle \left[\begin{matrix}\frac{l \cos{\left(q_{4} \right)} \dot{q}_{3}}{2} - \sin{\left(q_{3} + q_{4} \right)} \dot{q}_{1} + \cos{\left(q_{3} + q_{4} \right)} \dot{q}_{2}\\- \frac{l \cos{\left(q_{5} \right)} \dot{q}_{3}}{2} - \sin{\left(q_{3} + q_{5} \right)} \dot{q}_{1} + \cos{\left(q_{3} + q_{5} \right)} \dot{q}_{2}\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}- u_{1} \sin{\left(q_{3} + q_{4} \right)} + u_{2} \cos{\left(q_{3} + q_{4} \right)} + u_{3} \cos{\left(q_{4} \right)}\\- u_{1} \sin{\left(q_{3} + q_{5} \right)} + u_{2} \cos{\left(q_{3} + q_{5} \right)} - u_{3} \cos{\left(q_{5} \right)}\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}- \sin{\left(q_{3} + q_{4} \right)} & \cos{\left(q_{3} + q_{4} \right)}\\- \sin{\left(q_{3} + q_{5} \right)} & \cos{\left(q_{3} + q_{5} \right)}\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}- \cos{\left(q_{4} \right)} & 0 & 0\\\cos{\left(q_{5} \right)} & 0 & 0\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}0\\0\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}\frac{\frac{\cos{\left(q_{3} - q_{4} + q_{5} \right)}}{2} + \frac{\cos{\left(q_{3} + q_{4} - q_{5} \right)}}{2} + \cos{\left(q_{3} + q_{4} + q_{5} \right)}}{\sin{\left(q_{4} - q_{5} \right)}} & 0 & 0\\\frac{\frac{\sin{\left(q_{3} - q_{4} + q_{5} \right)}}{2} + \frac{\sin{\left(q_{3} + q_{4} - q_{5} \right)}}{2} + \sin{\left(q_{3} + q_{4} + q_{5} \right)}}{\sin{\left(q_{4} - q_{5} \right)}} & 0 & 0\end{matrix}\right]\end{split}\], \[\begin{split}\displaystyle \left[\begin{matrix}u_{1}\\u_{2}\end{matrix}\right] = \left[\begin{matrix}\frac{\left(\frac{\cos{\left(q_{3} - q_{4} + q_{5} \right)}}{2} + \frac{\cos{\left(q_{3} + q_{4} - q_{5} \right)}}{2} + \cos{\left(q_{3} + q_{4} + q_{5} \right)}\right) u_{3}}{\sin{\left(q_{4} - q_{5} \right)}}\\\frac{\left(\frac{\sin{\left(q_{3} - q_{4} + q_{5} \right)}}{2} + \frac{\sin{\left(q_{3} + q_{4} - q_{5} \right)}}{2} + \sin{\left(q_{3} + q_{4} + q_{5} \right)}\right) u_{3}}{\sin{\left(q_{4} - q_{5} \right)}}\end{matrix}\right]\end{split}\], a) two positions (or configurations) of car 2 relative to cars 1 and 3, b) 
