C++ Drake符号表达式计算异常问题及使用规范咨询

Great question! Transitioning from PyDrake to C++ Drake can be tricky because the C++ API enforces strict type checking (unlike PyDrake, which handles many implicit conversions behind the scenes). Let's break down each of your issues:

1. Why does 0.01*v.dot(v) fail but (0.01*v).dot(v) work? #

In C++, v (returned by NewContinuousVariables) is a VectorX<symbolic::Variable>. The key difference here lies in operator precedence and type promotion:

• When you write 0.01*v.dot(v), the compiler first evaluates v.dot(v) (member function calls have higher priority than multiplication), which returns a scalar symbolic::Expression. While mathematically valid, Eigen's template system sometimes struggles with implicit conversions between double and symbolic::Expression in this context, triggering the "cannot convert Expression to Variable" error.
• Wrapping 0.01*v in parentheses explicitly computes a VectorX<symbolic::Expression> first (multiplying a double by a Variable vector promotes elements to Expression). Calling .dot(v) on this vector then correctly returns a scalar symbolic::Expression, which matches the type expected by AddCost.

2. Why does transpose multiplication ((0.01*v).transpose()*v) fail? #

The issue here is the return type mismatch:

• (0.01*v).transpose() gives a RowVectorX<symbolic::Expression>, and multiplying this by v (a VectorX<symbolic::Variable>) results in a 1x1 MatrixX<symbolic::Expression>, not a scalar Expression.
• AddCost doesn't have an overload that accepts a 1x1 matrix directly. To fix this, extract the scalar value from the matrix:

  prog.AddCost(
      ((J_V_WS*v - V_WS).transpose()*(J_V_WS*v - V_WS))(0, 0) 
      + ((0.01*v).transpose()*v)(0, 0)
  );
  

  Alternatively, stick with the .dot() method, which directly returns a scalar Expression and avoids this matrix type issue.

3. Why does v.dot(v) fail but v.transpose()*v work? #

This again boils down to type handling quirks in Eigen and Drake's overloads:

• v.dot(v) for a VectorX<symbolic::Variable> returns a symbolic::Expression, but Eigen's template deduction may not properly resolve the overload to match AddCost's expectations.
• v.transpose()*v returns a 1x1 Matrix<symbolic::Expression>, and Drake's AddCost has an overload that automatically converts this 1x1 matrix to a scalar Expression, hence it works.

To make v.dot(v) work, explicitly promote v to a VectorX<symbolic::Expression> first:

prog.AddCost(Eigen::VectorX<drake::symbolic::Expression>(v).dot(v));

Official C++ Drake Symbolic Computation Guidelines #

Drake's official documentation includes detailed resources for symbolic computation:

• The Symbolic Computation chapter covers type rules, operator overloading, and best practices for working with symbolic::Variable and symbolic::Expression.
• The MathematicalProgram API docs specify exact parameter types for methods like AddCost and AddConstraint, which helps avoid type mismatch errors.

内容的提问来源于stack exchange，提问作者Chen Wang