| Summary: | The zero moment point (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>Z</mi> <mi>M</mi> <mi>P</mi> </mrow> </semantics> </math> </inline-formula>) and the linearized inverted pendulum model linking the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>Z</mi> <mi>M</mi> <mi>P</mi> </mrow> </semantics> </math> </inline-formula> to the center of gravity (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>O</mi> <mi>G</mi> </mrow> </semantics> </math> </inline-formula>) have an important role in the control of the postural equilibrium (balance) of biped robots and lower-limb exoskeletons. A solution for balance real time control, closing the loop from the joint actual values of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>O</mi> <mi>G</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>Z</mi> <mi>M</mi> <mi>P</mi> </mrow> </semantics> </math> </inline-formula>, has been proposed by Choi. However, this approach cannot be practically implemented: While the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>Z</mi> <mi>M</mi> <mi>P</mi> </mrow> </semantics> </math> </inline-formula> actual value is available from the center of pressure (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>o</mi> <mi>P</mi> </mrow> </semantics> </math> </inline-formula>) measured under the feet soles, the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>O</mi> <mi>G</mi> </mrow> </semantics> </math> </inline-formula> is not measurable, but it can only be indirectly assessed from the joint-angle measures, the knowledge of the kinematics, and the usually poorly known weight distribution of the links of the chain. Finally, the possible presence of unknown external disturbance forces and the nonlinear, complex nature of the kinematics perturb the simple relationship between the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>Z</mi> <mi>M</mi> <mi>P</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>O</mi> <mi>G</mi> </mrow> </semantics> </math> </inline-formula> in the linearized model. The aim of this paper is to offer, starting from Choi’s model, a practical implementation of closed-loop balance control fusing <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>o</mi> <mi>P</mi> </mrow> </semantics> </math> </inline-formula> and joint-angle measures, eliminating possible inconsistencies. In order to achieve this result, we introduce a model of the linearized inverted pendulum for an extended estimation, not only of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>O</mi> <mi>G</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>Z</mi> <mi>M</mi> <mi>P</mi> </mrow> </semantics> </math> </inline-formula>, but also of external disturbances. This model is then used, instead of Choi’s equations, for estimation and balance control, using <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">H</mi> <mo>∞</mo> </msub> </semantics> </math> </inline-formula> theory. As the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>O</mi> <mi>G</mi> </mrow> </semantics> </math> </inline-formula> information is recovered from the joint-angle measures, the identification of a statistically equivalent serial chain (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>E</mi> <mi>S</mi> <mi>C</mi> </mrow> </semantics> </math> </inline-formula>) linking the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>O</mi> <mi>G</mi> </mrow> </semantics> </math> </inline-formula> to the joint angles is also discussed.
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