Contents
ODE Solvers
Please see the ODE's users manual for general ODE documentation.
In general, rigid body simulators solve
- Kinematics constraints
- Collision and contact constraints
Rigid body dynamics
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ODE's constraint solver uses a full coordinate system approach and enforces joint and contact constraints as posed by the linear complementarity problem (LCP).
Basic Governing Equations of Constrained Dynamics
Before we discuss the solvers, here is a very brief note here on the governing dynamics equations. Simple Euler's discretization yields
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Constraints are described by the constraint Jacobian
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If we rewrite in matrix form we have:
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Substitute
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Left multiply top row of the matrix equation by
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ODE is semi-implicit in that the Jacobians
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Solvers
ODE ships with two default solvers
Dantzig's Agorithm dWorldStep()
- This algorithm will attempt to achieve a numerically exact solution. It is about one order of magnitude slower than SOR PGS LCP solver and its convergence behavior is less predictable in practice.
Successive Over-Relaxation (SOR) Projected Gauss-Seidel (PGS) LCP solver dWorldQuickStep()
- Essentially a Gauss-Seidel algorithm with solution vector projected into the allowable solution space at every update. The PR2 robot simulations default to this algorithm running at 1kHz (to match mechanism controller update rate of the real robot).
Dantzig's Agorithm
Please refer to step.cpp for implementation details. Various references contain discussions on this algorithm, see 2.7.1 in Michael Cline, "Rigid Body Simulation with Contacts and Constraints" for example. See also the Cottle and Dantzig book for details, Baraff extended the Dantzig algorithm to include friction in his SIGGRAPH 1994 paper. Also, chapter 14 of Murilo Coutinho's book "Guide to Dynamic Simulations of Rigid Bodies and Particle Systems" has detailed introduction to both Dantzig's algorithm and Baraff's friction extention.
The Dantzig algorithm solves general BLCP (Linear Complementarity Problem with Bounds), which has the form:
Solve:
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such that:
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In ODE's step.cpp,
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The Dantzig algorithm applies to more general BLCP. It incrementally computes intermediate solutions for each entry in the unknown vector:
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We first define the different sets based on properties of unknowns: Clamped Set
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Similarly, Non-Clamped Set
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or
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Do-not-care Set
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latex error! exitcode was 1 (signal 0), transscript follows: This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=latex) kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: /usr/share/texlive/texmf-dist/web2c/fmtutil.cnf mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /usr/share/httpd/.texlive2018/texmf-config/web2c/fmtutil.cnf /usr/bin/mktexfmt: mkdir(/usr/share/httpd/.texlive2018/) failed: Permission denied I can't find the format file `latex.fmt'!that satisfies:
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where
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During execution of Dantzig's algorithm, the left top
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Procedures of Dantzig's algorithm are: If we have only bounded constraints (bilateral constraints with lower and upper bounds), then all the indices are mapped to set
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Else if we have a mixture of unbounded and unbounded constraints, Dantzig algorithm does LDLT factorization and solve the first
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When we hit the first friction constraint, compute the corresponding lower and upper bound, using normal force at the same contact.
Assume
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Once we finish a complete loop on
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SOR PGS LCP
As implemented in ODE's quickstep.cpp, and reiterating the solution procedure from several popular literatures here.
We are essentially solving a system of linear equations where the solution space is non-negative in parts of the system.
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where based on the derivations from governing equations in the previous section,
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and
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If we solve for
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then it follows that
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for
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Formulate the desired solution in the form of acceleration1 (inverse mass matrix times constraint forces), denoted by
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then
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and
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for
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where each
latex error! exitcode was 1 (signal 0), transscript follows: This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=latex) kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: /usr/share/texlive/texmf-dist/web2c/fmtutil.cnf mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /usr/share/httpd/.texlive2018/texmf-config/web2c/fmtutil.cnf /usr/bin/mktexfmt: mkdir(/usr/share/httpd/.texlive2018/) failed: Permission denied I can't find the format file `latex.fmt'!is projected into its corresponding solution space depending on the type of constraint specified.
At every iteration, for each
latex error! exitcode was 1 (signal 0), transscript follows: This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=latex) kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: /usr/share/texlive/texmf-dist/web2c/fmtutil.cnf mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /usr/share/httpd/.texlive2018/texmf-config/web2c/fmtutil.cnf /usr/bin/mktexfmt: mkdir(/usr/share/httpd/.texlive2018/) failed: Permission denied I can't find the format file `latex.fmt'!update above, constraint accelerations
latex error! exitcode was 1 (signal 0), transscript follows: This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=latex) kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: /usr/share/texlive/texmf-dist/web2c/fmtutil.cnf mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /usr/share/httpd/.texlive2018/texmf-config/web2c/fmtutil.cnf /usr/bin/mktexfmt: mkdir(/usr/share/httpd/.texlive2018/) failed: Permission denied I can't find the format file `latex.fmt'!are updated in the following manner:
latex error! exitcode was 1 (signal 0), transscript follows: This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=latex) kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: /usr/share/texlive/texmf-dist/web2c/fmtutil.cnf mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /usr/share/httpd/.texlive2018/texmf-config/web2c/fmtutil.cnf /usr/bin/mktexfmt: mkdir(/usr/share/httpd/.texlive2018/) failed: Permission denied I can't find the format file `latex.fmt'!
for
latex error! exitcode was 1 (signal 0), transscript follows: This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=latex) kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: /usr/share/texlive/texmf-dist/web2c/fmtutil.cnf mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /usr/share/httpd/.texlive2018/texmf-config/web2c/fmtutil.cnf /usr/bin/mktexfmt: mkdir(/usr/share/httpd/.texlive2018/) failed: Permission denied I can't find the format file `latex.fmt'!
where
latex error! exitcode was 1 (signal 0), transscript follows: This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=latex) kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: /usr/share/texlive/texmf-dist/web2c/fmtutil.cnf mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /usr/share/httpd/.texlive2018/texmf-config/web2c/fmtutil.cnf /usr/bin/mktexfmt: mkdir(/usr/share/httpd/.texlive2018/) failed: Permission denied I can't find the format file `latex.fmt'!
For more details please see the list of references.
to clarify, in quickstep.cpp, $$\bar{a}_c$$ is denoted by variable fc as of svn revision 1675 (1)
